Concepts of Symmetry in the Work of Wolfgang Pauli Domenico Giulini Max-Planck-Institute for Gravitational Physics Albert-Einstein-Institute Am M¨ uhlenberg 1 D-14476 Golm, Germany February 29, 2008 Abstract “Symmetry” was one of the most important methodological themes in 20th-century physics and is probably going to play no lesser role in physics of the 21st century. As used today, there are a variety of interpretations of this term, which differ in meaning as well as their mathematical consequences. Symmetries of crystals, for example, gen- erally express a different kind of invariance than gauge symmetries, though in specific situations the distinctions may become quite subtle. I will review some of the various notions of “symmetry” and highlight some of their uses in specific examples taken from Pauli’s scientific œvre. This paper is based on a talk given at the conference Wolfgang Pauli’s Philosophical Ideas and Contemporary Science, May 20.-25. 2007, at Monte Verita, Ascona, Switzerland. 1
Transcript
Concepts of Symmetry
in the Work of Wolfgang Pauli
Domenico GiuliniMax-Planck-Institute for Gravitational Physics
Albert-Einstein-InstituteAm Muhlenberg 1
D-14476 Golm, Germany
February 29, 2008
Abstract
“Symmetry” was one of the most important methodological themesin 20th-century physics and is probably going to play no lesser rolein physics of the 21st century. As used today, there are a variety ofinterpretations of this term, which differ in meaning as well as theirmathematical consequences. Symmetries of crystals, for example, gen-erally express a different kind of invariance than gauge symmetries,though in specific situations the distinctions may become quite subtle.I will review some of the various notions of “symmetry” and highlightsome of their uses in specific examples taken from Pauli’s scientificœvre.
This paper is based on a talk given at the conference WolfgangPauli’s Philosophical Ideas and Contemporary Science, May 20.-25.2007, at Monte Verita, Ascona, Switzerland.
In the Introduction to Pauli’s Collected Scientific Papers, the editors, RalphKronig and Victor Weisskopf, make the following statement:
It is always hard to look for a leading principle in the work of agreat man, in particular if his work covers all fundamental prob-lems of physics. Pauli’s work has one common denominator: hisstriving for symmetry and invariance. [...] The tendency towardsinvariant formulations of physical laws, initiated by Einstein, hasbecome the style of theoretical physics in our days, upheld and de-veloped by Pauli during all his life by example, stimulation, andcriticism. For Pauli, the invariants in physics where the symbolsof ultimate truth which must be attained by penetrating throughthe accidental details of things. The search for symmetry andgeneral validity transcend the limits of physics in Pauli’s work;it penetrated his thinking and striving throughout all phases ofhis life, in all fields of philosophy and psychology.”([38], Vol. 1,p. viii)
Indeed, if I were asked to list those of Pauli’s scientific contributions whichmake essential use of symmetry concepts and applied group theory, I wouldcertainly include the following, which form a substantial part of Pauli’sscientific œvre:1
Relativity theory and Weyl’s extension thereof (1918-1921), theHydrogen atom in matrix mechanics (1925), exclusion principle(1925), anomalous Zeeman effect and electron spin (1925), non-relativistic wave-equation for spinning electron (1927), covari-ant QED (1928, Jordan), neutrino hypothesis (1930), Kaluza-Klein theory and its projective formulation (1933), theory of γ-matrices (1935), Poincare-invariant wave equations (1939, Fierz),general particle statistics and Lorentz invariance (1940, Belin-fante), spin-statistics (1940), once more General Relativity andKaluza-Klein theory (1943, Einstein), meson-nucleon interactionand differential geometry (1953), CPT theorem (1955), β-decayand conservation of lepton charge (‘Pauli group’, 1957), unifyingnon-linear spinor equation (collaboration with Heisenberg, 1957-58), group structure of elementary particles (1958, Touschek).
Amongst the theoretical physicists of his generation, Pauli was certainlyoutstanding in his clear grasp of mathematical notions and methods. He had1 Two of the listed themes, “meson-nucleon interaction and differential geometry” and
“unifying non-linear spinor equation”, were never published in scientific journals (inthe second case Heisenberg published for himself without Pauli’s consent) but can befollowed from his letters and manuscripts as presented in [45].
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a particularly sober judgement of their powers as well as their limitationsin applications to physics and other sciences. Let us once more cite Kronigand Weisskopf:
Pauli’s works are distinguished by their mathematical rigour andby a thorough and honest appraisal of the validity of assumptionsand conclusions. He was a true disciple of Sommerfeld in hisclear mathematical craftsmanship. By example and sharp criti-cism he constantly tried to maintain a similarly high standard inthe work of other theoretical physicists. He was often called theliving conscience of theoretical physicists. ([38], Vol. 1, p. viii)
It seems plausible that this critical impregnation dates back to his school-days, when young Pauli read, for example, Ernst Mach’s critical analysisof the historical development of the science of mechanics, a copy of whichPauli received as a present from his Godfather (Mach) at around the ageof fourteen. Mach’s “Mechanik”, as this book is commonly called, startsout with a discussion of Archimedes’ law of the lever, thereby criticisingthe following symmetry consideration ([43], p. 11-12): Imagine two equalmasses, M, and a perfectly stiff and homogeneous rod of length L, bothbeing immersed into a static homogeneous vertical gravitational field, wherethe rod is suspended at its midpoint, m, from a point p above; see Fig. 1.What happens if we attach the two equal masses to the ends of the rodand release them simultaneously without initial velocity? An immediatesymmetry argument suggests that it stays horizontal; it might be given asfollows: Everything just depends on the initial geometry and distribution ofmasses, which is preserved by a reflection at the plane perpendicular to therod through p andm. Suppose that after release the rod dropped at one sideof the suspension pointm, then the mirror image of that process would havethe same initial condition with the rod dropping to the other side. This is acontradiction if the laws governing the process are assumed to be reflectionsymmetric and deterministic (unique outcome for given initial condition).This argument seems rigorous and correct. Now, how does one get fromhere to the law of the lever? The argument criticised by Mach is as follows:Assume that the condition for equilibrium depends only on the amount ofmass and its suspension point on the rod, but not on its shape. Then wemay replace the mass to the left of m by two masses of half the amount eachon a small rod in equilibrium, as shown in the second (upper right) picture.Then replace the suspension of the small rod by two strings attached to theleft arm of the original rod, as shown in the third (lower left) picture, andobserve that the right one is just under the suspension point m, so that itdoes not disturb the equilibrium if it were cut away as in the last (lower right)picture of Fig. 1. The weak point in the argument is clearly the transitionfrom the second to the third picture: There is no global symmetry connectingthem, even though locally, i.e. regarding the small rod only, it connects two
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MM
m
p
=⇒
=⇒Figure 1: The law of the lever ‘derived’ from alleged symmetry consider-ations. The step from the upper right (second) to the lower left (third)picture does not follow. The small (blue) balls represent half the mass ofthe big (red) balls.
equilibrium positions. It is easy to see that, in fact, the assumption that aglobal equilibrium is maintained in this change is equivalent to Archimedes’law of the lever. This example shows (in admittedly a fairly trivial fashion)that alleged symmetry properties can work as a petitio principii for the lawto be derived. This is essentially the criticism of Mach.
The reason why we consider this ‘derivation’ of the law of the lever to bea petitio principii is that we have other, physically much more direct ways toactually derive it from dynamical first principles. From that point of view thealleged symmetry is to be regarded as an artifact of the particular law andcertainly not vice versa. The observed symmetry requires an explanationin terms of the dynamical laws, which themselves are to be established inan independent fashion. This is how we look upon, say, the symmetry ofcrystals or the symmetric shape of planetary orbits.
On the other hand, all fundamental dynamical theories of 20th cen-tury physics are motivated by symmetry requirements. They are commonlylooked at as particularly simple realisations of the symmetries in question,given certain a priori assumptions. It is clear that, compared to the previousexample, there are different concepts of symmetry invoked here. However,there also seems to be a shift in attitude towards a more abstract under-standing of ‘physical laws’ in general.
What makes Pauli an interesting figure in this context is that this shiftin attitude can be traced in his own writings. Consider Special Relativity asan example, thereby neglecting gravity. One may ask: What is the generalrelation between the particular symmetry (encoded by the Poincare group)of spacetime and that very same symmetry of the fundamental interactions
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(weak, strong, and electromagnetic, but not gravity)? Is one to be consideredas logically prior to the other? For example, if we take Einstein’s originaloperationalist attitude, we would say that the geometry of spacetime isdefined through the behaviour of ‘rods’ and ‘clocks’, which eventually shouldbe thought of as physical systems obeying the fundamental dynamical laws.In fact, Einstein often complained about the fact that rods and clocks areintroduced as if they were logically independent of the dynamical laws, e.g.,in a discussion remark at the 86th meeting of the Gesellschaft DeutscherNaturforscher und Arzte in Bad Nauheim in 1920:2
It is a logical shortcoming of the Theory of Relativity in itspresent form to be forced to introduce measuring rods and clocksseparately instead of being able to construct them as solutions todifferential equations. ([66], Vol. 7, Doc. 46, p. 353)
From that viewpoint, symmetry properties of spacetime are nothing but aneffective codification of the symmetries of the fundamental laws. Conse-quences like ‘length contraction’ and ‘time dilation’ in Special Relativity arethen only effectively described as due to the geometry of spacetime, whereasa fundamental explanation clearly has to refer to the dynamical laws thatgovern clocks and rods. This was clearly the attitude taken by H.A. Lorentzand H.Poincare, though in their case still somehow afflicted with the ideaof a material æther that, in principle, defines a preferred rest frame, so thatthe apparent validity of the principle of relativity must be interpreted as dueto a ‘dynamical conspiracy’.3 In his famous article on Relativity for the En-cyclopedia of Mathematical Sciences, the young Pauli proposes to maintainthis view, albeit without the idea on a material æther. He writes:4
Should one, then, in view of the above remarks, completely aban-don any attempt to explain the Lorentz contraction atomistically?We think that the answer to this question should be No. The con-traction of a measuring rod is not an elementary but a very com-plicated process. It would not take place except for the covariance
2 German original: “Es ist eine logische Schwache der Relativitatstheorie in ihrem heuti-gen Zustande, daß sie Maßstabe und Uhren gesondert einfuhren muß, statt sie alsoLosungen von Differentialgleichungen konstruieren zu konnen.”
3 H.A. Lorentz still expressed this viewpoint well after the formulation of Special Rela-tivity, for example in [41], p. 23.
4 German original: “Ist aber das Bestreben, die Lorentz-Kontraktion atomistisch zu ver-stehen, vollkommen zu verwerfen? Wir glauben diese Frage verneinen zu mussen. DieKontraktion des Maßstabes ist kein elementarer, sondern ein sehr verwickelter Prozeß.Sie wurde nicht eintreten, wenn nicht schon die Grundgleichungen der Elektronentheo-rie sowie die uns noch unbekannten Gesetze, welche den Zusammenhalt des Elektronsselbst bestimmen, gegenuber der Lorentz-Gruppe kovariant waren. Wir mussen ebenpostulieren, daß dies der Fall ist, wissen aber auch, daß dann, wenn dies zutrifft, dieTheorie imstande sein wird, das Verhalten von bewegten Maßstaben und Uhren atom-istisch zu erklaren.” ([58], p. 30.)
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with respect to the Lorentz group of the basic equations of elec-tron theory, as well as those laws, as yet unknown to us, whichdetermine the cohesion of the electron itself. We can only postu-late that this is so, knowing that then the theory will be capable ofexplaining atomistically the behaviour of moving measuring rodsand clocks.” ([54], p. 15.)
Very recently, this traditional view has once more been defended under thename of ‘Physical Relativity’ [7] against todays more popular view, accord-ing to which Special Relativity is about the symmetry properties of space-time itself. Clearly, the latter view only makes sense if spacetime is endowedwith its own ontological status, independently of the presence of rods andclocks.
This shift in emphasis towards a more abstract point of view is also re-flected in Pauli’s writings, for example in the Preface to the English editionof his ‘Theory of Relativity’ of 1956, where the abstract group-theoreticproperties of dynamical laws are given an autonomous status in the expla-nation of phenomena:
The concept of the state of motion of the ‘luminiferous æther’,as the hypothetical medium was called earlier, had to be given up,not only because it turned out to be unobservable, but because itbecame superfluous as an element of a mathematical formalism,the group-theoretical properties of which would only be disturbedby it. By the widening of the transformation group in generalrelativity the idea of a distinguished inertial coordinate systemcould also be eliminated by Einstein, being inconsistent with thegroup-theoretical properties of the theory.
Pushed to an extreme, this attitude results in the belief that the mostfundamental laws of physics are nothing but realisations of basic symme-tries. Usually this is further qualified by adding that these realisations arethe most ‘simple’ ones, at least with respect to some intuitive measure ofsimplicity. Such statements are well known from Einstein’s later scientificperiod and also from Heisenberg in connection with his ‘unified theory’ ofelementary particles, for which he proposed a single non-linear differentialequation, whose structure was almost entirely motivated by its symmetryproperties. Heisenberg made this point quite explicitly in his talk entitledPlanck’s discovery and the foundational issues of atomism5, delivered dur-ing the celebrations of Max Planck’s 100th anniversary—at which occasionWolfgang Pauli received the Max-Planck medal in absentia—, where he also5 German original: “Die Plancksche Entdeckung und die philosophischen Grundfragen
der Atomlehre”.
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talked about his own ‘unified theory’:6
The mentioned equation contains, next to the three natural units[c, h, l], merely mathematical symmetry requirements. Theserequirements seem to determine everything else. In fact, oneshould just regard this equation as a particularly simple repre-sentation of the symmetry requirements, which form the actualcore of the theory.
Pauli, who briefly collaborated with Heisenberg on this project, did not atall share Heisenberg’s optimism that a consistent quantum-field theory couldbe based on Heisenberg’s non-linear field equation. His objections concernedseveral serious technical aspects, overlayed with an increasing overall dislikeof Heisenberg’s readiness to make premature claims, particularly when madepublicly.
However, I think it is fair to say that the overall attitude regarding theheuristic role and power of symmetry principles in fundamental physics,expressed by Heisenberg in the above quote, was also to a large extentshared by Pauli, not only in his later scientific life. This is particularlytrue for symmetry induced conservation laws, towards which Pauli had verystrong feelings indeed. Examples from his later years will be discussed inlater sections (e.g. Sect. 3.7). An example from his early scientific life ishis strong resistance against giving up energy-momentum conservation forindividual elementary processes, while keeping it on the statistical average.Such ideas were advocated in the “new radiation theory” of Bohr, Kramers,and Slater of early 1924 [5] and again by Bohr in connection with β-decay,which Pauli called spiritual somersaults in a letter to Max Delbruck. A weekafter his famous letter suggesting the existence of the neutrino, Pauli wroteto Oskar Klein in a letter dated Dec. 12th 1930:7
6 German original: “Die erwahnte Gleichung enthalt neben den drei naturlichen Maßein-heiten nur noch mathematische Symmetrieforderungen. Durch diese Forderungenscheint alles weitere bestimmt zu sein. Man muß eigentlich die Gleichung nur als einebesonders einfache Darstellung der Symmetrieforderungen, aber diese Forderung als deneigentlichen Kern der Theorie betrachten.” ([45], Vol. IV, Part IVB, p. 1168)
7 German original: “Erstens scheint es mir, daß der Erhaltungssatz fur Energie-Impulsdem fur die Ladung doch sehr weitgehend analog ist und ich kann keinen theoretischenGrund dafur sehen, warum letzterer noch gelten sollte (wie wir es ja empirisch furden β-Zerfall wissen), wenn ersterer versagt. Zweitens mußte bei einer Verletzung desEnergiesatzes auch mit dem Gewicht etwas sehr merkwurdiges passieren. [...] Dieswiderstrebt meinem physikalischen Gefuhl auf das außerste! Denn es muß dann sogarauch fur das Gravitationsfeld, das von dem ganzen Kasten (samt seinem radioaktivenInhalt) selber erzeugt wird (...), angenommen werden, daß es sich andern kann, wahrendwegen der Erhaltung der Ladung das nach außen erzeugte elektrostatische Feld (beideFelder scheinen mir doch analog zu sein; das wirst Du ja ubrigens auch aus deinerfunfdimensionalen Vergangenheit noch wissen) unverandert bleiben soll.” ([45], Vol. II,Doc. [261], p. 45-46)
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First it seems to me, that the conservation law for energy-momentum is largely analogous to that for electric charge, and Icannot see a theoretical reason why the latter should still be valid(as we know empirically from β-decay) if the former fails. Sec-ondly, something strange should happen to the weight if energyconservation fails. [...] This contradicts my physical intuitionto an extreme! For then one has to even assume that the grav-itational field produced [...] by the box (including the radioac-tive content) can change, whereas the electrostatic field must re-main unchanged due to charge conservation (both fields seem tome analogous; as you will remember from your five-dimensionalpast).
This is a truly remarkable statement. Not many physicists would nowadaysdare suggesting such an intimate connection between the conservation lawsof charge and energy-momentum. What Pauli hints at with his last remarksin brackets is the Kaluza-Klein picture, in which electric charge is inter-preted as momentum in an additional space dimension in a five-dimensionalspacetime.
It is not difficult to find explicit commitments from Pauli’s later scientificlife expressing his belief in the heuristic power of symmetry considerations.Let me just select two of them. The first is from his introduction to theInternational Congress of Philosophers, held in Zurich in 1954, where Paulistates:8
“It seems likely to me, that the reach of the mathematical groupconcept in physics is not yet fully exploited.”
The second is from his closing remarks as the president of the conference“50 Years of Relativity” held in Berne in 1955, where with respect to thestill unsolved problem of whether and how the gravitational field should bedescribed in the framework of Quantum-Field-Theory he remarks:9
It seems to me, that the heart of the matter [the problem of quan-tising the gravitational field] is not so much the linearity or non-linearity, but rather the fact that there is present a more generalgroup than the Lorentz group.
This, in fact, implicitly relates to much of the present-day research that isconcerned with that difficult problem.8 German original: “Es ist mir wahrscheinlich, dass die Tragweite des mathematischen
Gruppenbegriffes in der Physik heute noch nicht ausgeschopft ist.” ([38], Vol. 2, p. 1345)9 German original: “Es scheint mir also, daß nicht so sehr die Linearitat oder Nichtlin-
earitat der Kern der Sache ist, sondern eher der Umstand, daß hier eine allgemeinereGruppe als die Lorentzgruppe vorhanden ist.”([38], Vol. 2, p. 1306)
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Before we can discuss specific aspects of ‘symmetry’ in Pauli’s work inSection 3, we wish to recall various aspects of symmetry principles as usedin physics.
2 Remarks on the notion of symmetry
2.1 Spacetime
The term ‘symmetry’ is used in such a variety of meanings, even in physics,that it seems appropriate to recall some of its its main aspects. One aspectis that which mathematicians call an ‘automorphism’ and which basicallymeans a ‘structure preserving self-map’. Take as an example (conceptuallynot an easy one) the modern notion of spacetime. First of all it is a set,M, the members of which are events, or better, ‘potential events’, since wedo not want to assume that every spacetime point to be an actual physicalevent in the sense that a material happening is taking place, or at leastnot one which is dynamically relevant to the problem at hand.10 That setis endowed with certain structures which are usually motivated throughoperational relations of actual physical events.
One such structure could be that of a preferred set of paths, which repre-sent inertial (i.e. force free) motions of ‘test bodies’, that is, localised objectswhich do not react back onto spacetime structure. This defines a so-called‘path-structure’ (compare [12][10]), which in the simplest case reduces to anaffine structure in which the preferred paths behave, intuitively speaking,like ‘straight lines’. This can clearly be said in a much more precise form(see, e.g., [60]). Under very mild technical assumptions (not even involvingcontinuity) one may then show that the only automorphisms of that ‘iner-tial structure’ can already be narrowed down to the inhomogeneous Galileior Lorentz groups, possibly supplemented by constant scale transformations(cf. [24][28]).11
Another structure to start with could have been that of a causal relationon M. That is, a partial order relation which determines the pairs of pointson spacetime which, in principle, could influence each other in form of apropagation process based on ordinary matter or light signals. The auto-morphism group of that structure is then the subgroup of bijections on Mthat, together with their inverse, preserve this order relation. For example,in case of Minkowski space, where the causal relation is determined by the10 Minkowski was well aware that empty domains of spacetime may cause conceptual
problems. Therefore, in his famous 1908 Cologne address Space and Time (Germanoriginal: “Raum und Zeit”), he said: In order to not leave a yawning void, we wishto imagine that at every place and at every time something perceivable exists. Germanoriginal: “Um nirgends eine gahnende Leere zu lassen, wollen wir uns vorstellen, daßallerorten und zu jeder Zeit etwas Wahrnehmbares vorhanden ist”. ([47], p. 2)
11 We shall from now on use ‘Poincare group’ for ‘inhomogeneous Lorentz group’ and‘Lorentz group’ for ‘homogeneous Lorentz group’.
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light-cone structure, it may be shown that the most general automorphismis given by a Poincare transformation plus a constant rescaling[1][73]. Since,according to Klein’s Erlanger Programm [36], any geometry may be charac-terised by its automorphism group, the geometry of Minkowski space is, upto constant rescalings, entirely encoded in the causal relations.
The same result can be arrived at through topological considerations.Observers (idealised to be extensionless) move in spacetime on timelikecurves. Take the set C of all (not necessarily smooth) timelike curves whichare continuous in the standard (Euclidean) topology TE of Minkowski space-time M. Now endow M with a new topology, TP, called the path topology,which is the finest topology on M which induces the same topology on eachpath in C as the standard (Euclidean) topology TE. The new topology TP
is strictly finer than TE and has the following remarkable property: Theautomorphism group of (M, TP)12, i.e. the group of bijections of M which,together with their inverses, preserve TP, is just the Poincare group extendedby the constant rescalings [32]. This is possibly the closest operational mean-ing one could attribute to the topology of spacetime, since in TP a set inspacetime is open if and only if every observer “times” it to be open.
All this is meant to illustrate that there are apparently different waysto endow spacetime with structures that are, physically speaking, more orless well motivated and which lead to the same automorphism group. Thatgroup may then be called the group of spacetime symmetries. So far, thisgroup seems to bear no direct relation to any dynamical law. However, thephysical meaning of such statements of symmetry is tight to an ontologicalstatus of spacetime points. We assumed from the onset that spacetime isa set M. Now, recall that Georg Cantor, in his first article on transfiniteset-theory [8], started out with the following definition of a set:13
By a ‘set’ we understand any gathering-together M of deter-mined well-distinguished objects m of our intuition or of ourthinking (which are called the ‘elements’ of M) into a whole.
Hence we may ask: Is a point in spacetime, a ‘potential event’ as we calledit earlier, a “determined well-distinguished object of our intuition or of ourthinking”? This question is justified even though modern axiomatic set the-ory is more restrictive in what may be called a set (for otherwise it runsinto the infamous antinomies) and also stands back from any characterisa-tion of elements in order to not confuse the axioms themselves with their12 In the standard topological way of speaking this is just the ‘homeomorphism group’ of
(M, TP).13 German original: “Unter einer ‘Menge’ verstehen wir jede ZusammenfassungM von bes-
timmten wohlunterschiedenen Objecten m unserer Anschauung oder unseres Denkens(welche die ‘Elemente’ von M genannt werden) zu einem Ganzen.” ([8], p. 481)
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possible interpretations.14 However, applications to physics require inter-preted axioms, where it remains true that elements of sets are thought ofas definite as in Cantors original definition. But it is just this definitenessthat seems to be physically unwarranted in application to spacetime. Themodern general-relativistic viewpoint takes that into account by a quotientconstruction, admitting only those statements as physically meaningful thatare invariant under the group of (differentiable) permutations of spacetimepoints. This is possible only because all other structures on spacetime, inparticular the metric and with it the causal structure, are not fixed once andfor all but are subsumed into the dynamical fields. Hence no non-dynamicalbackground structures remain, except those that are inherent in the defini-tion of a differentiable manifold. The group of automorphisms is thereforethe whole diffeomorphism group of spacetime, which, in some sense, comessufficiently close to the group of all permutations.15
2.2 Dynamical symmetries versus covariance
What is the relation between spacetime automorphisms and symmetriesof dynamical laws? Before we can answer this, we have to recall what asymmetry of a dynamical law is.
For definiteness, let us restrict attention to dynamical laws in classical(i.e. non-quantum) physics. The equations of motion generally take theform of systems of differential equations, which we here abbreviate with EM(Equation of Motion). These equations involve two types of quantities: 1)background structures, collectively abbreviated here by Σ, and 2) dynamicalentities, collectively abbreviated here by Φ. The former will typically berepresented by geometric objects on M (tensor fields, connections, etc),which are taken from a somehow specified set B of ‘admissible backgrounds’.Typical background structures are external sources, like currents, and thegeometry of spacetime in non-general-relativistic field theories. Dynamicalentities typically involve ‘particles’ and ‘fields’, which in the simplest casesare represented by maps to and from spacetime,
γ : R → M (‘particle’) , (1a)ψ : M → V (‘field’) , (1b)
14 This urge for a clean distinction between the axioms and their possible interpretationsis contained in the famous and amusing dictum, attributed to David Hilbert by hisstudent Otto Blumenthal: “One must always be able to say ’tables’, ‘chairs’, and ‘beermugs’ instead of ’points, ‘lines’, and ‘planes”. (German original: “Man muß jederzeit anStelle von ’Punkten’, ‘Geraden’ und ‘Ebenen’ ’Tische’, ‘Stuhle’ und ‘Bierseidel’ sagenkonnen.”)
15 There are clearly much more general bijections of spacetime than continuous or evendifferentiable ones. However, the diffeomorphism group is still n-point transitive,that is, given any two n-tuples of mutually distinct spacetime points, (p1, · · · , pn)
and (q1, · · · , qn), there is a diffeomorphism φ such that for all 1 ≤ i ≤ n we haveΦ(pi) = qi; this is true for all positive integers n.
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and were V is usually some vector space.In order to state the equations of motion, one has to first specify a set of
so-called16kinematically possible trajectories out of which the dynami-cal entities Φ are taken and solutions to the equations of motion are sought.Usually this involves particle trajectories which are sufficiently smooth (typ-ically piecewise twice continuously differentiable) and fields which are suf-ficiently smooth and in addition have a sufficiently rapid fall-off at largespatial distances, so as to give rise to finite quantities of energy, angular-momentum, etc. This space of kinematically possible trajectories will bedenoted by K. According to the discussion above, the equation of motiontakes two arguments, one from B the other from K, and is hence written inthe form
EMΣ | Φ = 0 , (2)
where the zero on the right-hand side may be a many-component object.Equation (2) should be read as a selection criterion on the set K, dependingon the externally specified values of Σ. We shall sometimes write EMΣ forEMΣ | · to denote the particular equation of motion for Φ correspondingto the choice Σ for the background structures. In general, the sets of solu-tions to (2) for variable Σ are Σ-dependent subset DΣ ⊂ K, whose elementsare called the dynamically possible trajectories16. We can now saymore precisely what is usually meant by a symmetry:
Definition 1 An abstract group G is called a symmetry group of theequations of motion iff17 the following conditions are satisfied:
1. There is an effective (see below) action G×K → K of G on the set ofkinematically possible trajectories, denoted by (g,Φ) 7→ g ·Φ.
2. This action leaves the subset DΣ ⊂ K invariant; that is, for all g in Gwe have:
EMΣ | Φ = 0 ⇐⇒ EMΣ | g ·Φ = 0 . (3)
Recall that an action is called effective if no group element other than thegroup identity fixes all points of the set it acts on. Effectiveness is requiredin order to prevent mathematically trivial and physically meaningless exten-sions of G. What really matters are the orbits of G in K, that is, the subsetsOΦ = g · Φ | g ∈ G for each Φ ∈ K. If the action were not effective, wecould simply reduce G to a smaller group with an effective action and thesame orbits in K, namely the quotient group G/G ′, where G ′ is the normalsubgroup of elements that fix all points of K.
It should be noted that this definition is still very general due to thefact that no further condition is imposed on the action of G, apart from the16 This terminology is due to James Anderson [2].17 Throughout we use “iff” as abbreviation for “if and only if”.
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obvious one of effectivity. For example, for fields one usually requires theaction to be ‘local’, in the sense that for any point p of spacetime, the value(g · ψ)(p) of the g-transformed field should be determined by the value ofthe original field at some point p ′ of spacetime, and possibly finitely manyderivatives of ψ at p ′. If there are no dependencies on the derivatives, theaction is sometimes called ‘ultralocal’. Note that the point p ′ need not beidentical to p, but it is assumed to be uniquely determined by g and p. Astriking example of what can happen if locality is not imposed is given bythe vacuum Maxwell equations (no external currents), which clearly admitthe Poincare group as ultralocally acting symmetry group. What is less wellknown is the fact that they also admit the inhomogeneous Galilei group assymmetry group18, albeit the action is non-local; see [20] or Chap. 5.9 of[21]. (There are also other non-local symmetries of the vacuum Maxwellequations [19].)
To be strictly distinguished from the notion of symmetry is the notionof covariance, which we define as follows:
Definition 2 An abstract group G is called a covariance group of theequations of motion iff the following conditions are satisfied:
1. There is an effective action G×K → K of G on the set of kinematicallypossible trajectories, denoted by (g,Φ) 7→ g ·Φ.
2. There is also an action (this time not necessarily effective) G×B → Bof G on the set of background structures, likewise denoted by (g, Σ) 7→g · Σ.
3. The solution-function Σ 7→ DΣ ⊂ K from B into the subsets of K isG-equivariant. This means the following: If g · DΣ denotes the setg ·Φ | Φ ∈ DΣ, then, for all g in G, we have
g · DΣ = Dg·Σ . (4)
An alternative way to say this is that the relation that EM establisheson B × K via (2) is G invariant, that is, for all g in G, we have
EMΣ | Φ = 0 ⇐⇒ EMg · Σ | g ·Φ = 0 . (5)
The obvious difference between (3) and (5) is that in the former case thebackground structure is not allowed to change. The transformed dynamicalentity is required to satisfy the very same equation as the untransformed one,18 This is different from, and certainly more surprising than, the better known (ultra local)
Galilei symmetry of Maxwell’s equations in the presence of appropriate constitutiverelations between the electric field ~E and the electric displacement-field ~D on one side,and between the magnetic induction-field ~B and the magnetic field ~H on the other; seee.g. [39] and [27].
14
whereas for a covariance it is only required to satisfy a suitably changed setof equations. Here ‘changed’ refers to the fact that g ·Σ is generally differentfrom Σ. Hence it is clear that a symmetry group is automatically also acovariance group, by just letting it act trivially on the set B of backgroundstructures. The precise partial converse is as follows: Given a covariancegroupG with action on B, then for each Σ ∈ B define the ‘stabiliser subgroup’of Σ in G as the set of elements in G that fix Σ,
StabG(Σ) := g ∈ G | g · Σ = Σ . (6)
Then the subgroup StabG(Σ) of the covariance group is also a symmetrygroup of the equation of motion EMΣ.
The requirement of covariance is a rather trivial one, since it can alwaysbe met by suitably taking into account all the background structures anda sufficiently general action of G on B. To see how this works in a specificexample, consider the ordinary ‘heat equation’ for the temperature field T(κ is a dimensionful constant):
∂tT − κ∆T = 0 . (7)
Let G = E3 × R be the 7-parameter group of Euclidean motions (rotationsand translations in R3) and time translations, whose defining representationon spacetime (R3 × R) is denoted by g → ρg, then G acts effectively onthe set of temperature fields via g · T := T ρg−1 (the inverse being justintroduced to make this a left action). It is immediate from the structure of(7) that this implements G as symmetry group of this equation. The back-ground structures implicit in (7) are: a) a preferred split of spacetime intospace and time, 2) a preferred measure and orientation of time, and c) a pre-ferred distance measure on space. There are many ways to parametrise thisstructure, depending on the level of generality one starts from. If, for exam-ple, we start from Special Relativity, we only list those structural elementsthat we need on top of the Minkowski metric ηµν = diag(1,−1,−1,−1) inorder to write down (7). They are given by a single constant and normalisedtimelike vector field n, by means of which we can write (7) in the form
EMn | T := nµ∂µT − κ(nµnν − ηµν)∂µ∂νT = 0 . (8)
In the special class of inertial reference frames in which nµ = (1, 0, 0, 0)
equation (8) reduces to (7). From the structure of (8) it is obvious that thisequation admits the whole Poincare group of Special Relativity as covariancegroup. However, the symmetry group it contains is the stabiliser subgroupof the given background structure. The latter is given by the vector fieldn, whose stabiliser subgroup within the Poincare group is just E3 × R, thesame as for (7).
Had we started from a higher level of generality, in which no preferredcoordinate systems are given to us as in Special Relativity, we would write
15
the heat equation in the form
EMn, g | T := nµ∇µT − κ(nµnν − gµν)∇µ∇νT = 0 , (9)
where now n as well as g feature as background structures. n is again spec-ified as unit timelike covariant-constant vector field, g as a flat metric, and∇ as the unique covariant derivative operator associated to g (i.e. torsionfree and preserving g). Since ∇ is here taken as a unique function of g,it does not count as independent background structure. Once again it isclear from the structure of (9) that the covariance group is now the wholediffeomorphism group of spacetime. However, the symmetry group remainsthe same as before since the stabiliser subgroup of the pair (g, n) is E3 ×R.
This example should make clear how easy it is to almost arbitrarily inflatecovariance groups by starting from higher and higher levels of generalityand adding the corresponding extra structures into ones list of backgroundstructures. This possibility is neither surprising nor particularly disturbing.Slightly more disturbing is the fact that a similar game can be played withsymmetries, at least on a very formal level. The basic idea is to simplydeclare background structures to be dynamical ones by letting their values bedetermined by equations. We may do this since we have so far not qualified‘equations of motion’ as any special sort of equations. For example, in thespecial relativistic context we may just take (8) and let n be determined by
nµnνηµν = 1 , ∂µnν = 0 . (10)
Then (8) and (10) together define a background free (from the special rela-tivistic point of view) system of equations for T, n which has the full Poincaregroup as symmetry group. Its symbolic form is
EM ∅ | T, n = 0 , (11)
where the 0 on the right-hand side has now 18 components: one for (8), onefor the first equation in (10), and 16 (= 4 × 4) for the second equation in(10). But note that its T -sector of solution space is not the same as thatof (7), as it now also contains solutions for different n. However, as theequations (10) for n do not involve T , the total solution space for (n, T) canbe thought of as fibred over the space of allowed n, with each fibre over nbeing given by the solutions T of (8) for that given n. Each such fibre is afaithful image of the original solution space of (7), suitably transformed bya Lorentz boost that relates the original n in (7) (i.e. nµ = (1, 0, 0, 0)) tothe chosen one.
Even more radically, we could take (9) and declare n and g to be dy-namical entities obeying the extra equations
nµnνgµν = 1 , ∇µnν = 0 , Riem[g] = 0 , (12)
16
where Riem is the Riemann curvature tensor of g, so that the last equationin (12) just expresses flatness of g. The system consisting of (9) and (12)has no background structures and admits the full diffeomorphism group assymmetry group. It is of the symbolic form
EM ∅ | T, n, g = 0 , (13)
which now comprises 36 components: the 16 as above and an additionalset of 20 for the independent components of Riem. Again, note that theT -sector of solution space of (9) is now much bigger that of (7) of or (8).With any solution T it also contains its diffeomorphism-transformed one,T ′ = T φ−1, where φ ∈ Diff(M). Again, since the equations for n and gdo not involve T , the total solution space is fibred over the allowed n andg fields, with each fibre corresponding to a faithful image of the originalsolution space for (7).
Finally we remark that, in principle, constants appearing in equationsof motion could also be addressed as background structures whose valuesmight eventually be determined by more general dynamical theories. Forexample, one might speculate (as was done some time ago in the so-calledBrans-Dicke theories) that the gravitational constant is actually the value ofsome field that only in the present epoch of our Universe has settled to a spa-tially constant and quasi-static value, but whose value at much earlier timeswas significantly different. Another example from Quantum Field Theoryconcerns the idea that masses of elementary particles are dynamically gen-erated by the so-called Higgs field (whose existence is strongly believed butnot yet experimentally confirmed).
In any case, the important message from the considerations of this sub-section is the following: symmetries emerge or disappear if, respectively,background structures become dynamical (Σ → Φ) or dynamical structures‘freeze’ (Φ → Σ).
2.3 Observable versus gauge symmetries
Within the concept of symmetry as explained so far, an important distinc-tion must be made between observable symmetries on one hand, and gaugesymmetries on the other. An observable symmetry transforms a state ora history of states (trajectory) into a different, that is, physically distin-guishable state or history of states. On the other hand, a gauge symmetrytransforms a state or a history of states into a physically indistinguishablestate or a history of states. In this case there is a redundancy in the math-ematical description, so that the map from mathematical labels to physicalstates is not faithful. This is usually associated with a group, called thegroup of gauge transformations, denoted by Ggau, which acts on the set ofstate labels such that two such labels correspond to the same physical stateiff they lie in the same orbit of Ggau.
17
It is clear that the notion of ‘distinguishability’ introduced here refersto the set of observables, i.e. functions on state space that are physicallyrealisable in the widest sense. Assuming for the moment that this was welldefined, we could attempt a definition as follows:
Definition 3 Let G be a symmetry group in the sense of Definition 1. Theng ∈ G is called an observable or physical symmetry iff there exists aΦ ∈ DΣ and a physical observable that separates g ·Φ from Φ. If no suchobservable exists, g is called a gauge symmetry.
It is clear that for a theoretician the stipulation of what functions on statespace correspond to physically realisable observables is itself of hypotheticalnature. However, what is important for us at this point is merely thatrelative to such a stipulation the distinction between observables and gaugesymmetries makes sense. In the mathematical practice gauge symmetries areoften signalled by an underdeterminedness of the equations of motion, whichsometimes simply fail to restrict the motion in certain degrees of freedomwhich are then called ‘gauge degrees of freedom’. In that case, given anysolution Φ ∈ DΣ, we can obtain another solution, Φ ′, by just changing Φ inthose non-determined degrees of freedom in an arbitrary way. For example,if the equations of motion are obtained via an action principle, such spuriousdegrees of freedom will typically reveal their nature through the propertythat motions in them are not associated with any action. As a result, theequations of motion, which are just the condition for the stationarity ofthe action, will not constrain the motion in these directions. Conversely,if according to the action principle the motion in some degree of freedomcosts action, it can hardly be called a redundant one. In this sense an actionprinciple is not merely a device for generating equations of motion, but alsocontains some information about observables.
The combination of observable and gauge symmetries into the total sym-metry group G need not at all be just that of a semi-direct or even directproduct. Often, in field theory, the gauge group Ggau is indeed a subgroupof G, in fact an invariant (normal) one, but the observable symmetries, Gobs,are merely a quotient and not a subgroup of G. In standard group theoreticterms one says that G is a Ggau−extension of Gobs. This typically happensin electromagnetism or more generally in Yang-Mills type gauge theories orGeneral Relativity with globally charged configurations. In this case onlythe ‘gauge transformations’ with sufficiently rapid fall-off at large spatialdistances are proper gauge transformations in our sense, whereas the longranging ones cost action if performed in real time19 and therefore have tobe interpreted as elements of Gobs; see e.g. [23] and Chap. 6 of [33].
This ends our small excursion into the realm of meanings of ‘symmetry’.We now turn to the discussion of specific aspects in Pauli’s work.19 By the very definition of global charge, which is just the derivative of the action with
respect to a long-ranging ’gauge transformation’.
18
3 Specific comments on symmetries in Pauli’swork
The usage of symmetry concepts in Pauli’s work is so rich and so diverse thatit seems absolutely hopeless, and also inappropriate, to try to present themin a homogeneous fashion with any claim of completeness. Rather, I willcomment on various subjectively selected aspects without in any way sayingthat other aspects are of any lesser significance. In fact, I will not includesome of his most outstanding contributions, like, for example, the formula-tion of the exclusion principle, the neutrino hypothesis, or his anticipationof Yang-Mills Gauge Theory for the strong interaction. There exist excellentreviews and discussions of these topics in the literature. Specifically I wishto refer to Bartel van der Waerden’s contribution Exclusion Principle andSpin to the Pauli Memorial Volume ([18], pp. 199-244), Norbert Straumann’srecent lecture on the history of the exclusion principle [68], Pauli’s own ac-count of the history of the neutrino (in English: [57], pp. 193-217; in German:[38], Vol. 2, pp. 1313-1337 and [55], p. 156-180), Chien-Shiung Wu’s accountThe Neutrino in the Pauli Memorial Volume ([18], pp. 249-303), and the his-torical account of gauge theories by Lochlainn O’Raifeartaigh and NorbertStraumann [50]. A non-technical overview concerning Pauli’s Belief in Ex-act Symmetries is given by Karl von Meyenn [46]. Last, but clearly notleast, I wish to mention Charles Enz’s fairly recent comprehensive scientificbiography [16] of Wolfgang Pauli, which gives a detailed discussion of hisscientific œvre.
In this contribution I rather wish to concentrate on some particular as-pects of the notion of symmetry that are directly related to the foregoingdiscussion in Sections 2.2 and 2.3, as I feel that they are somewhat neglectedin the standard discussions of symmetry.
3.1 The hydrogen atom in matrix mechanics
In January 1926 Pauli managed to deduce the energy spectrum for the Hy-drogen atom from the rules of matrix mechanics. For this he implicitlyused the fact that the mechanical problem of a point charge moving in aspherically symmetric force-field with a fall-off proportional to the squareof the inverse distance has a symmetry group twice as large (i.e. of twicethe dimension) as the group of spatial rotations alone, which it contains.Hence the total symmetry group is made half of a ‘kinematical’ part, refer-ring to space, and half of a ‘dynamical’ part, referring to the specific forcelaw (1/r2 fall-off). Their combination is a proper physical symmetry groupthat transforms physically distinguishable states into each other. In thegiven quantum-mechanical context one also speaks of ‘spectrum generating’symmetries.
Let us recall the classical problem in order to convey some idea where the
19
symmetries and their associated conserved quantities show up, and how theymay be employed to solve the dynamical problem. Consider a mass-point ofmass m and position coordinate ~r in the force field ~F(~r) = −(K/r2)~n, wherer is the length of ~r, ~n := ~r/r, and K is some dimensionful constant. Then,according to Newton’s 3rd law (an overdot stands for the time derivative),
~r = −k
r2~n (k = K/m) . (14)
Next to energy, there are three obvious conserved quantities correspond-ing to the three components of the angular-momentum vector (here writtenper unit mass)
~ = ~r× ~r . (15)
But there are three more conserved quantities, corresponding to the compo-nents of the following vector (today called the Lenz-Runge vector),
~e = k−1~r× ~ − ~n . (16)
Conservation can be easily verified by differentiation of (16) using (14) and~n = ~ × ~n/r2. Hence on has (` = length of ~)
from which the classical orbit immediately follows: Setting ~r · ~e = re cosϕ,the last equation (17) reads
r =`2/k
1+ e cosϕ, (18)
which is the well known equation for a conic section in the plane perpendic-ular to ~, focus at the origin, eccentricity e (= length of ~e), and latus rectum2`2/k. The vector ~e points from the origin to the point of closest approach(periapsis). The few steps leading to this conclusion illustrate the powerbehind the method of working with conservation laws which, in turn, restson an effective exploitment of symmetries.
The total energy per unit mass is given by E = 12~r2 − k/r. A simple
calculation shows thate2 − 1 = 2E`2/k2 , (19)
which allows to express E as function of the invariants e2 and `2. This is therelation which Pauli shows to have an appropriate matrix analogue, whereit allows to express the energy in terms of the eigenvalues of the matrices for`2 and e2 which Pauli determines, leading straight to the Balmer formula.
From a modern point of view one would say that, for fixed energy E < 0,20
the state space of this problem carries a Hamiltonian action of the Lie algebra20 For E > 0 one obtains a Hamiltonian action of so(1,3).
20
so(4), generated by the 3+3 quantities ~ and ~e. Quantisation then consistsin the problem to represent this Lie algebra as a commutator algebra ofself-adjoint operators and the determination of spectra of certain elementsin the enveloping algebra. This is what Pauli did, from a modern point ofview, but clearly did not realise at the time. In particular, even though hecalculated the commutation relations for the six quantities ~ and ~e, he didnot realise that they formed the Lie algebra for so(4), as he frankly statedmuch later (1955) in his address on the occasion of Hermann Weyl’s 70thbirthday:21
Similarly I did not know that the matrices which I had derivedfrom the new quantum mechanics in order to calculate the en-ergy values of the hydrogen atom were a representation of the4-dimensional orthogonal group.
This may be seen as evidence for Pauli’s superior instinct for detecting rel-evant mathematical structures in physics. Much later, in a CERN-report of1956, Pauli returned to the representation-theoretic side of this problem [53].
3.2 Particles as representations of spacetime automorphisms
The first big impact of group theory proper on physics took place in quan-tum theory, notably through the work or Eugene Wigner [72] and HermannWeyl [70]. While in atomic spectroscopy the usage of group theory couldbe looked upon merely as powerful mathematical tool, it definitely acquireda more fundamental flavour in (quantum) field theory. According to a dic-tum usually attributed to Wigner, every elementary system (particle) inspecial-relativistic quantum theory corresponds to a unitary irreducible rep-resentation of the Poincare group.22 In fact, all the Poincare invariant linearwave equations on which special-relativistic quantum theory is based, knownby the names of Klein & Gordon, Weyl, Dirac, Maxwell, Proca, Rarita &Schwinger, Bargmann & Wigner, Pauli & Fierz, can be understood as projec-tion conditions that isolate an irreducible sub-representation of the Poincaregroup23 within a reducible one that is easy to write down. More concretely,the latter is usually obtained as follows: Take a field ψ on spacetime M with21 German original: “Ebensowenig wußte ich, daß die Matrices, die ich ausgerechnet hatte,
um die Energiewerte des Wasserstoffatoms aus der neuen Quantenmechanik abzuleiten,eine Darstellung der 4-dimensionalen othogonalen Gruppe gewesen sind”. ([45], Vol. IV,Part III, Doc. [2183], p. 402) Note that, in modern terminology, Pauli actually refers toa representation of the Lie algebra of the orthogonal group.
22 The converse is not true, since there exist unitary irreducible representations whichcannot correspond to (real) particles, for example the so-called ‘tachyonic’ ones, corre-sponding to spacelike four-momenta.
23 More precisely, its universal cover R4 o SL(2,C), or sometimes an extension thereof bythe discrete transformations of space and time reversal.
21
values in a finite-dimensional complex vector space V . Let D be a finite-dimensional irreducible representation of the (double cover of the) Lorentzgroup SL(2,C) on V .24 It is uniquely labelled by a pair (p, q) of two positiveinteger- or half-integer-valued numbers. In the standard terminology, 2pcorresponds to the number of unprimed, 2q to the number of primed spinorindices of ψ. The set of such fields furnishes a linear representation of the(double cover of the) Poincare group, R4 o SL(2,C), where the action of thegroup element g = (a,A) is given by
g ·ψ := D(A)(ψ g−1) , (20)
or for the Fourier transform ψ,
g · ψ := exp(ipµaµ)D(A)(ψ A−1) . (21)
One immediately infers from (21) that irreducibility implies that ψ musthave support on a single SL(2,C) orbit in momentum space. Here one usuallyrestricts to those orbits consisting of non-spacelike p (those with spacelikep give rise to the tachyonic representations which are deemed unphysical),which are labelled by pµp
µ = m2 with non-negative m. For ψ this meansthat it obeys the Klein-Gordon equation ( +m2)ψ = 0. This is alreadyhalf the way to an irreducible representation, insofar as it now contains onlymodes of fixed mass. But these modes still contains several spins up to themaximal value p + q. A second and last step then consists of projectingout one (usually the highest) spin, which gives rise to the equations namedabove. In this fashion the physical meanings of mass and spin merge withthe abstract mathematical meaning of mere labels of irreducible representa-tions. Mass and spin are the most elementary attributes of physical objects,so that objects with no other attributes are therefore considered elementary.As just described, these elementary attributes derive from the representa-tion theory of a group whose significance is usually taken to be that it isthe automorphism group of spacetime. However, as already discussed inSections 1 and 2.1, this point of view presupposes a hierarchy of physicalthinking in which spacetime (here Minkowski space) is considered an entityprior to (i.e. more fundamental than) matter, which may well be challenged.A more consistent but also more abstract point of view would be to thinkof the abstract25 Poincare group as prior to the matter content as well asthe spacetime structure and to derive both simultaneously. Here ‘deriving’24 The representation D is never unitary, simply because the Lorentz group has no non-
trivial finite-dimensional unitary irreducible representations. But it will give rise to aninfinite-dimensional representation on the linear space of fields ψ which will indeed beunitary.
25 ‘Abstract’ here means to consider the isomorphicity class of the group as mathematicalstructure, without any interpretation in terms of transformations of an underlying setof objects.
22
a spacetime structure (geometry) from a group would be meant in the senseof Klein’s Erlanger Programm [36].
We have already discussed in Section 1 Pauli’s shift in emphasis towardsa more abstract point of view as regards spacetime structure. But also asregards to matter he was, next to Wigner, one of the proponents to put sym-metry considerations first and to derive the wave equations of fundamentalfields as outlined above. Based on previous work by Fierz on the theory offree wave equations for higher spin [17], Fierz and Pauli published their veryinfluential paper On Relativistic Equations for Particles of Arbitrary Spinin an Electromagnetic Field ([38], Vol. 2, pp. 873-894) which is still muchcited today.
In fact, much earlier, in his 1927 paper Quantum Mechanics of theMagnetic Electron26, Pauli succeeded to implement the electron’s spin intonon-special-relativistic quantum mechanics in an entirely representation-theoretic fashion as regards the (Lie algebra of) spatial rotations. In contrastto the other (translational) degrees or freedom, spin does not appear as thequantisation of an already existent classical degree of freedom. This musthave appeared particularly appealing to Pauli, who never wanted the elec-tron’s ‘spin’ to be understood as an intrinsic angular momentum due to aspatial rotation of a material structure. When Pauli introduced the new spinquantum-number for the electron in his 1924 paper On the Influence of theVelocity Dependence of the Electron Mass on the Zeeman Effect27 he delib-erately stayed away from any model interpretation and cautiously referredto it as a peculiar, classically indescribable disposition of two-valuedness ofthe quantum-theoretic properties of the light-electron28. At that time anunderstandable general scepticism against possible erroneous prejudices im-posed by the usage of classical models had already firmly established itselfin Pauli’s (and others) thinking.
As much justified as this is in view of Quantum Mechanics, this had alsoled to overstatements to the effect that spin has no classical counterpartand that any classical model is even classically contradictory in the senseof violating Special Relativity. As regards the second point, which was alsopushed by Pauli, we refer to the detailed discussion in [22]. To the first pointwe first wish to mention that composite models with half-integer angularmomentum states exist in ordinary Quantum Mechanics (without spin), as,e.g., pointed out by Bopp & Haag in 1950 [6]. This is possible if their classicalconfiguration space contains the whole group SO(3) of spatial rotations.26 German original: “Zur Quantenmechanik des magnetischen Elektrons”. ([38], Vol. 2,
pp. 306-330)27 German original: “Uber den Einfluß der Geschwindigkeitsabhangigkeit der Elektronen-
masse auf den Zeemaneffekt”. ([38], Vol. 2, pp. 201-213)28 German original: “eine eigentumliche, klassisch nicht beschreibbare Art von Zwei-
deutigkeit der quantentheoretischen Eigenschaften des Leuchtelektrons” ([38], Vol. 2,p. 213).
23
Pauli himself showed in his 1939 paper On a Criterion for Single- or Double-Valuedness of the Eigenfunctions in Wave Mechanics29 the possibility ofdouble-valued wavefunctions, which are the ones that give rise to half-integerangular momentum states. Moreover, in classical mechanics there is also aprecise analog of Wigner’s notion of an elementary system. Recall thatthe space of states of a mechanical system is a symplectic manifold (phasespace). The analog of an irreducible and unitary representation of the groupof spacetime automorphisms is now a transitive and Hamiltonian action ofthis group on the symplectic manifold. It is interesting to note that thisclassical notion of an elementary system was only formulated much laterthan, and in the closest possible analogy with, the quantum mechanicalone. An early reference where this is spelled out is [4]. The classificationof elementary systems is now equivalent to the classification of symplecticmanifolds admitting such an action. An early reference where this has beendone is [3]. Here, as expected, an intrinsic angular momentum shows upas naturally as it does in Quantum Mechanics. What makes it slightlyunusual (but by no means awkward or even inconsistent) is the fact that itcorresponds to a phase space30 that is not the cotangent bundle (space ofmomenta) over some configuration space of positions.
Pauli’s later writings also show this strong inclination to set the funda-mentals of (quantum) field theory in group-theoretic terms. In his surveyRelativistic Field Theories of Elementary Particles ([38], Vol. 2, pp. 923-952), written for the 1939 Solvay Congress, Pauli immediately starts a dis-cussion of “transformation properties of the field equations and conservationlaws”. His posthumously published notes on Continuous Groups in Quan-tum Mechanics [53] focus exclusively on Lie-algebra methods in representa-tion theory.
Today we are used to define physical quantities like energy, momentum,and angular momentum as the conserved quantities associated to spacetimeautomorphisms via Noether’s theorem. Here, too, Pauli was definitely anearly advocate of this way of thinking. Reviews on the subject writtenshortly after Pauli’s death show clear traces of Pauli’s approach; see e.g. [35].
3.3 Spin and statistics
Pauli’s proof of the spin-statistics correlation [51] (also [38], Vol. 2, pp. 911-922), first shown by Markus Fierz in his habilitation thesis [17], is a trulyimpressive example for the force of abstract symmetry principles. Here wewish to recall the basic lemmas on which it rests, which merely have to dowith classical fields and representation theory.29 German original: “Uber ein Kriterium fur Ein- oder Zweiwertigkeit der Eigenfunktionen
in der Wellenmechanik”. ([38], Vol. 2, pp. 847-868)30 The phase space for classical spin is a 2-sphere, which is compact and therefore leads
to a finite-dimensional Hilbert space upon quantisation.
24
We begin by replacing the proper orthochronous Lorentz group by itsdouble (= universal) cover SL(2,C) in order to include half-integer spinfields. We stress that everything that follows merely requires the invarianceunder this group. No requirements concerning invariance under space- ortime reversal are needed!
We recall from the previous section that any finite-dimensional complexrepresentation of SL(2,C) is labelled by an ordered pair (p, q), where p and qmay assume independently all non-negative integer or half-integer values. 2pand 2q correspond to the numbers of ‘unprimed’ and ‘primed’ spinor indices,respectively. The tensor product of two such representations decomposes asfollows:
D(p,q) ⊗D(p ′,q ′) =
p+p ′⊕r=|p−p ′|
q+q ′⊕s=|q−q ′|
D(r,s) , (22)
where—and this is the important point in what follows—the sums proceedin integer steps in r and s. With each D(p,q) let us associate a ‘Pauli Index’,given by
π : D(p,q) → ((−1)2p , (−1)2q) ∈ Z2 × Z2 . (23)
This association may be extended to sums of such D(p,q) proceeding ininteger steps, simply by assigning to the sum the Pauli Index of its terms(which are all the same). Then we have31
According to their representations, we can associate a Pauli Index withspinors and tensors. For example, a tensor of odd/even degree has PauliIndex (−,−)/(+,+). The partial derivative, ∂, counts as a tensor of degreeone. Now consider the most general linear (non interacting) field equationsfor integer spin (here and in what follows
∑(· · · ) simply stands for “sum of
terms of the general form (· · · )”):∑∂(−,−)Ψ(+,+) =
∑Ψ(−,−) ,∑
∂(−,−)Ψ(−,−) =∑
Ψ(+,+) .(25)
These are invariant under
Θ :
Ψ(+,+)(x) 7→ + Ψ(+,+)(−x),
Ψ(−,−)(x) 7→ − Ψ(−,−)(−x) .(26)
Next consider any current that is a polynomial in the fields and their deriva-tives:
31 This may be expressed by saying that the map π is a homomorphism of semigroups. Onesemigroup consists of direct sums of irreducible representations proceeding in integersteps with operation ⊗, the other is Z2 × Z2, which is actually a group.
25
Then one has(ΘJ)(x) = −J(−x) . (28)
This shows that for any solution of the field equations with charge Q forthe conserved current J (Q being the space integral over J0) there is anothersolution (the Θ transformed) with charge −Q. It follows that charges ofconserved currents cannot be sign-definite in any SL(2,C)-invariant theoryof non-interacting integer spin fields. In the same fashion one shows thatconserved quantities, stemming from divergenceless symmetric tensors ofrank two, bilinear in fields, cannot be sign-definite in any SL(2,C) invarianttheory of non-interacting half-integer spin fields. In particular, the conservedquantity in question could be energy!
An immediate but far reaching first conclusion (not explicitly drawn byPauli) is that there cannot exist a relativistic generalisation of Schrodinger’sone- particle wave equation. For example, for integer-spin particles, onesimply cannot construct a non-negative spatial probability distribution de-rived from conserved four-currents. This provides a general argument forthe need of second quantisation, which in textbooks is usually restricted tothe spin-zero case.
Upon second quantisation the celebrated spin-statistics connection forfree fields can now be derived in a few lines. It says that integer spin fieldscannot be quantised using anti-commutators and half-integer spin field can-not be quantised using commutators. Here the so-called Jordan-Pauli dis-tribution plays a crucial role32 in the (anti)commutation relations, whichensures causality (observables localised in spacelike separated regions com-mute). Also, the crucial hypothesis of the existence of an SL(2,C) invariantstable vacuum state is adopted. Pauli ends his paper by saying:
In conclusion we wish to state, that according to our opinion theconnection between spin and statistics is one of the most impor-tant applications of the special relativity theory. ([51], p. 722)
It took almost 20 years before first attempts were made to generalise thisresult to the physically relevant case of interacting fields by Luders & Zu-mino [42].32 The Jordan-Pauli distribution was introduced by Jordan and Pauli in their 1927 paper
Quantum Electrodynamics of Uncharged Fields (“Zur Quantenelektrodynamik ladungs-freier Felder”; [38], Vol. 2, pp. 331-353) in an attempt to formulate Quantum Electro-dynamics in a manifest Poincare invariant fashion. It is uniquely characterised (up to aconstant factor) by the following requirements: (1) it must be Poincare invariant undersimultaneous transformations of both arguments; (2) it vanishes for spacelike separatedarguments; (3) it satisfies the Klein-Gordon equation. The (anti)commutators of thefree fields must be proportional to the Jordan-Pauli distribution, or to finitely manyderivatives of it.
26
3.4 The meaning of ‘general covariance’
General covariance is usually presented as the characteristic feature of Gen-eral Relativity. The attempted meaning is that a generally covariant lawtakes the ‘same form’ in all spacetime coordinate systems. However, in or-der to define the ‘form’ of a law one needs to make precisely the distinctionbetween background entities, which are constitutive elements of the law, andthe dynamical quantities which are to be obey the laws so defined (cf. Sec-tion 2.1). In the language we introduced above, ‘general covariance’ cannotjust mean simple covariance under all smooth and invertible transforma-tions of spacetime points, i.e. that the spacetime diffeomorphism group is acovariance group is the sense of Definition 2, for that would be easily achiev-able without putting any restriction on the intended law proper, as was al-ready pointed out by Erich Kretschmann in 1917 [37]. Einstein agreed withthat criticism of Kretschmann’s, which he called “acute” (German original:“scharfsinnig”)([66], Vol. 7, Doc. 4, pp. 38-41), and withdrew to the view thatthe principle of general covariance has at least some heuristic power in thefollowing sense:33
Between two theoretical systems which are compatible with expe-rience, that one is to be preferred which is the simpler and moretransparent one from the standpoint of the absolute differentialcalculus. Try to bring Newton’s gravitational mechanics in theform of generally covariant equations (four dimensional) and onewill surely be convinced that principle a)34 is, if not theoretically,but practically excluded.
But the principle of general covariance is intended as a non-trivial selec-tion criterion. Hence modern writers often characterise it as the requirementof diffeomorphism invariance, i.e. that the diffeomorphism group of space-time is a symmetry group in the sense of Definition 1. But then, as wehave seen above, the principle is open to trivialisations if one allows back-ground structures to become formally dynamical. This possibility can onlybe inhibited if one limits the amount of structure that may be added to the33 German original: “Von zwei mit der Erfahrung vereinbaren theoretischen Systemen
wird dasjenige zu bevorzugen sein, welches vom Standpunkte des absoluten Differen-tialkalkuls das einfachere und durchsichtigere ist. Man bringe einmal die NewtonscheGravitationsmechanik in die Form von kovarianten Gleichungen (vierdimensional) undman wird sicherlich uberzeugt sein, daß das Prinzip a) diese Theorie zwar nicht theo-retisch, aber praktisch ausschließt.” ([66], Vol. 7, Doc. 4, p. 39)
34 Einstein formulates principle a) thus: “Principle of relativity: The laws of nature exclu-sively contain statements about spacetime coincidences; therefore they find their naturalexpression in generally covariant equations.” ([66], Vol. 7, Doc. 4, p. 38)
27
dynamical fields.35
The reason why I mention all this here is that Pauli’s Relativity article is,to my knowledge, the only one that seems to address that point, albeit notas explicitly as one might wish. After mentioning Kretschmann’s objection,he remarks (the emphases are Pauli’s):36
The generally covariant formulation of the physical laws acquiresa physical content only through the principle of equivalence, inconsequence of which gravitation is described solely by the gik
and the latter are not given independently from matter, but arethemselves determined by the field equations. Only for this reasoncan the gik be described as physical quantities. ([54], p. 150)
Note how perceptive Pauli addresses the two central issues: 1) that onehas to limit the the amount of dynamical variables and 2) that dynamicalstructures have to legitimate themselves as physical quantities through theirback reaction onto other (matter) structures. It is by far the best few-line account of the issue that I know of, though perhaps a little hard tounderstand without the more detailed discussion given above in Section 2.2.Most modern textbooks do not even address the problem. See [25] for morediscussion.
3.5 General covariance and antimatter
In this section I wish to give a brief but illustrative example from Pauli’swork for the non-trivial distinction between observable physical symmetrieson one hand, and gauge symmetries on the other (cf. Section 2.3). Theexample I have chosen concerns an argument within the (now outdated)attempts to understand elementary particles as regular solutions of classicalfield equations. Pauli reviewed such attempts in a rather detailed fashion inhis Relativity article, with particular emphasis on Weyl’s theory, to whichhe had actively contributed in two of his first three published papers in 1919.
The argument proper says that in any ‘generally covariant’37 theory,which allows for regular static solutions representing charged particles, there35 Physically speaking, one may be tempted to just disallow such formal ‘equations of
motions’ whose solution space is (up to gauge equivalence) zero dimensional. But thiswould mean that one would have to first understand the solution space of a given theorybefore one can decide on its ‘general covariance’ properties, which would presumablyrender it a practically fairly useless criterion.
36 German original: Einen physikalischen Inhalt bekommt die allgemein kovariante For-mulierung der Naturgesetze erst durch das Aquivalenzprinzip, welches zur Folge hat, daßdie Gravitation durch die gik allein beschrieben wird, und daß diese nicht unabhangigvon der Materie gegeben, sondern selbst durch Feldgleichungen bestimmt sind. Erst de-shalb konnen die gik als physikalische Zustandsgroßen bezeichnet werden. ([58], p. 181)
37 Here ‘general covariance’ is taken to mean that the diffeomorphism group of spacetimeacts as symmetry group.
28
exists for any solution with mass m and charge e another such solution withthe same mass but opposite charge −e. Pauli’s proof looks like an almosttrivial application of general covariance and runs as follows: Let gµν(xλ) andAµ(xλ) represent the gravitational and electromagnetic field respectively.The hypothesis of staticity implies that coordinates (and gauges for Aµ)can be chosen such that all fields are independent of the time coordinate,x0, and that g0i ≡ 0 as well as Ai ≡ 0 for i = 1, 2, 3.38 Now considerthe orientation-reversing diffeomorphism φ : (x0,~x) 7→ (−x0,~x). It mapsthe gravitational field to itself while reversing the sign of A0 and hence ofthe electric field. General covariance assures these new fields to be againsolutions with the same total mass but opposite total electric charge.
Pauli presents this argument in his second paper addressing Weyl’stheory, entitled To the Theory of Gravitation and Electricity by HermannWeyl39 ([38], Vol. 2, pp. 13-23, here p. 18) and also towards the end of Sec-tion 67 of his Relativity article. The idea of this proof is due to Weyl whocommunicated it (without formulae) in his first two letters to Pauli ([45],Vol. 1, Doc. [1] and [2]), as Pauli also acknowledges in his paper ([38], Vol. 2,p. 18, footnote 2).
It is interesting to note that Einstein rediscovered the very same argu-ment in 1925 and found it worthy of a separate communication [13]. Atthe time it was common to all, Weyl, Pauli, and Einstein, to regard the ar-gument a nuisance and of essentially destructive nature. This was becauseat this time antiparticles had not yet been discovered so that the apparentasymmetry as regards the sign of the electric charges of fundamental par-ticles was believed to be a fundamental property of Nature. Already in hisfirst paper on Weyl’s theory ([38], Vol. 2, pp. 1-9), entitled Perihelion Motionof Mercury and Deflection of Rays in Weyl’s Theory of Gravitation40, Pauliemphasised:41
The main difficulty [with Weyl’s theory] is – apart from Ein-stein’s objection, which appears to me not yet sufficiently dis-proved – that the theory cannot account for the asymmetry be-tween the two sorts of electricity.
Now, there is an interesting conceptual point hidden in this argumentthat relates to our discussions in Sections 2.2 and 2.3. First of all, the two38 The latter conditions distinguish staticity from mere stationarity. The condition on Ai
may, in fact, be relaxed.39 German original: “Zur Theorie der Gravitation und der Elektrizitat von Hermann
Weyl”.40 German original: “Merkurperihelbewegung und Strahlenablenkung in Weyls Gravita-
tionstheorie”.41 German original: “Die Hauptschwierigkeit ist – neben Einstein’s Einwand, der mir dur-
chaus noch nicht hinreichend widerlegt scheint –, daß die Theorie von der Asymmetrieder beiden Elektrizitatsarten nicht befriedigend Rechenschaft zu geben vermag.” ([38],Vol. 2, p. 8)
29
solutions are clearly considered physically distinct, otherwise the argumentcould not be understood as contradicting the charge asymmetry in Nature.Hence the diffeomorphism involved cannot be considered a gauge trans-formation but rather corresponds to a proper physical symmetry. On theother hand, we know that diffeomorphisms within bounded regions must beconsidered as gauge transformations, for otherwise one would run into thedilemma set by the so-called “hole argument”42. Hence one faces the prob-lem of how one should characterise those diffeomorphisms which are not tobe considered as gauge transformations (cf. Section 2.3). It is conceivablethat this question is not decidable without contextual information. (See e.g.[23] and Chapter 6 of [33] for more discussion of this point.) The historicalsources have almost nothing to say about this, though there are suggestionsby all three mentioned authors how to circumvent the argument by addingmore non-dynamical structures, as a result of which general covariance islost. Einstein, being most explicit here, suggested the existence of a time-like vector field which fixes a time orientation. At least the so-defined timeorientation would then have to be considered as non-dynamical structureof type Σ (cf. Section 2.2) in order to break the symmetry group downto the stabiliser group of Σ. The time-orientation-reversing transformationused above would then not be a symmetry anymore. Similar suggestionswere made by Weyl, who also hinted at a structure to distinguish past andfuture:43
Their essential difference [of past and future] I take, contrary tomost physicists, to be a fact of much more fundamental meaningthan the essential difference between positive and negative charge.
In the last (5th) edition of Raum Zeit Materie, Hermann Weyl writes re-42 Let Ω be a bounded region in spacetime which is disjoint from a spacelike hypersurfaceΣ. Consider two solutions to the field equations which merely differ by the action of adiffeomorphism φ with support in Ω. If they are considered distinct, then the theorycannot have a well posed initial-value problem, since then for any Σ distinct solutionsexist with identical data on Σ. This is a rephrasing of Einstein’s original argument ([66],Vol. 4, Doc. 25, p. 574, Doc. 26, p 580, Vol. 6, Doc. 2, p. 10), which did not construct acontradiction to the existence of a well posed initial-value problem, but rather to therequirement that the gravitational field be determined by the matter content (moreprecisely: its energy momentum tensor). But this requirement is clearly never fulfilledin any generally covariant theory in which the gravitational field has its own degreesof freedom, independent of whether one regards diffeomorphisms as gauge. Slightlylater he rephrased it so as to construct a contradiction to the existence of a well posedboundary-value problem ([66], Vol. 6, Doc. 9, p. 110), which is also not the right thingto require from equations that describe the propagation of fields with own degrees offreedom.
43 German original: “Ihren Wesensunterschied [von Vergangenheit und Zukunft] halte ich,im Gegensatz zu den meisten Physikern, fur eine Tatsache von noch viel fundamentalererBedeutung als der Wesensunterschied zwischen positiver und negativer Elektrizitat.”([45], Vol. 1, Doc. [2], p. 6)
30
garding his unified theory (the emphases are Weyl’s): 44
The theory gives no clue as regards the disparity of positiveand negative electricity. But that cannot be taken as a reproachagainst the theory. For that disparity is based without doubt onthe fact that of both fundamental constituents of matter, the elec-tron and the hydrogen nucleus, the positively charged one is tightto another mass then the negatively charged one; it originatesfrom the nature of matter and not of the field.
Given that Weyl is talking about his unified field-theory of gravity andelectricity, whose original claim was to explain all of matter by means offield theory, this statement seems rather surprising. It may be taken as asign of Weyl’s beginning retreat from his once so ambitious programme.
3.6 Missed opportunities
3.6.1 Supersymmetry
One issue that attracted much attention during the 1960s was, whether theobserved particle multiplets could be understood on the basis of an all em-bracing symmetry principle that would combine the Poincare group withthe internal symmetry groups displayed by the multiplet structures. Thiscombination of groups should be non-trivial, i.e., not be a direct product, forotherwise the internal symmetries would commute with the spacetime sym-metries and lead to multiplets degenerate in mass and spin (see, e.g., [49]).Subsequently, a number of no-go theorems appeared, which culminated inthe now most famous theorem of Coleman & Mandula [11]. This theoremstates that those generators of symmetries of the S-matrix belonging to thePoincare group necessarily commute with those belonging to internal sym-metries. The theorem is based on a series of assumptions45 involving thecrucial technical condition that the S-matrix depends analytically on stan-dard scattering parameters. What is less visible here is that the structure ofthe Poincare group enters in a decisive way. This result would not follow for44 German original: “Die Theorie gibt keinen Aufschluß uber die Ungleichartigkeit von
positiver und negativer Elektrizitat. Das kann ihr aber nicht zum Vorwurf gemachtwerden. Denn jene Ungleichartigkeit beruht ohne Zweifel darauf, daß von den beidenUrbestandteilender der Materie, Elektron und Wasserstoffkern, der positiv geladene miteiner anderen Masse verbunden ist als der negaiv geladene; sie entspringt aus der Naturder Materie und nicht des Feldes.” ([71], p. 308)
45 The assumptions are: (1) there exists a non-trivial (i.e., 6= 1) S-matrix which dependsanalytically on s (the squared centre-of-mass energy) and t (the squared momentumtransfer); (2) the mass spectrum of one-particle states consists of (possibly infinite)isolated points with only finite degeneracies; (3) the generators (of the Lie algebra) ofsymmetries of the S-matrix contains (as a Lie-sub algebra) the Poincare generators;(4) some technical assumptions concerning the possibility of writing the symmetry gen-erators as integral operators in momentum space.
31
the Galilean group, as was explicitly pointed out by Coleman & Mandula([11], p. 159).
One way to avoid the theorem of Coleman & Mandula is to generalisethe notion of symmetries. An early attempt was made by Golfand & Likht-man [29], who constructed what is now known as a Super-Lie algebra, whichgeneralises the concept of Lie algebra (i.e. symmetry generators obeying cer-tain commutation relations) to one also involving anti-commutators. In thisway it became possible for the first time to link particles of integer and half-integer spin by a symmetry principle. It is true that Supersymmetry stillmaintains the degeneracy in masses and hence cannot account for the massdifferences in multiplets. But its most convincing property, the symmetrybetween bosons and fermions, suggested a most elegant resolution of thenotorious ultraviolet divergences that beset Quantum Field Theory.
It is remarkable that the idea of a cancellation of bosonic and fermioniccontributions to the vacuum energy density occurred to Pauli. In his lecturesSelected Topics in Field Quantization, delivered in 1950-51 (in print againsince 2000, [59]), he posed the question
..whether these zero-point energies [from Bosons and Fermions]can compensate each other. ([59], p. 33)
He tried to answer this question by writing down the formal expression forthe zero point energy density of a quantum field of spin j and mass mj > 0
(Pauli restricted attention to spin 0 and spin 1/2, but the generalisation isimmediate):
4π2Ej
V= (−1)2j(2j+ 1)
∫dkk2
√k2 +m2 . (29)
Cancellation should take place for high values of k. The expansion
4
∫K
0dkk2
√k2 +m2 = K4 +m2
jK2 −m4
j log(2K/mj) +O(K−1) (30)
shows that the quartic, quadratic, and logarithmic terms must cancel in thesum over j for the limit K → ∞ to exist. This implies that for n = 0, 2, 4
one must have∑j
(−1)2j(2j+ 1)mnj = 0 and
∑j
(−1)2j(2j+ 1) log(mj) = 0 . (31)
Pauli comments that
these requirements are so extensive that it is rather improbablethat they are satisfied in reality. ([59], p. 33)
Unless enforced by an underlying symmetry, one is tempted to add! Thiswould have been the first call for a supersymmetry in the year 1951.
32
However, the real world does not seem to be as simple as that. Super-symmetry, if at all existent, is strongly broken in the phase we live in. So farno supersymmetric partner of any existing particle has been detected, eventhough some of them (e.g., the neutralino) are currently suggested to be vi-able candidates for the missing-mass problem in cosmology. Future findings(or non-findings) at the Large Hadron Collider (LHC) will probably have adecisive impact on the future of the idea of supersymmetry, which—whetheror not it is realised in Nature—is certainly very attractive; and Pauli cameclose to it.
3.6.2 Kaluza-Klein Monopoles
Ever since its first formulation in 1921, Pauli as well as Einstein were muchattracted by the geometric idea of Theodor Kaluza and its refinement byOskar Klein, according to which the classical theories of the gravitationaland the electromagnetic field could be unified into a single theory, in whichthe unified field has the same meaning as Einsteins gravitational field inGeneral Relativity, namely as metric tensor of spacetime, but now in fiveinstead of four dimensions. The momentum of a particle in the additionalfifth direction (which is spacelike) is now to be interpreted as its charge.Charge is conserved because the geometry of spacetime is a priori restrictedto be independent of that fifth direction. The combined field equationsare exactly the five-dimensional analog of Einstein’s equations for GeneralRelativity.
A natural question to address in this unified classical theory was whetherit admits solutions that could represent particle-like objects. More precisely,the solution should be stationary, everywhere regular, and possess long-ranging gravitational and electromagnetic fields (usually associated withaspects of mass and charge). Pauli, who was very well familiar with thistheory since its first appearance46, kept an active interest in it even afterthe formulations of Quantum Mechanics and early Quantum Electrodynam-ics, which made it unquestionable for him that the problem of matter couldnot be adequately addressed in the framework of a classical field theory,unlike Einstein, who maintained such a hope in various forms until the endof his life in 1955.
It is therefore remarkable that in 1943 Einstein and Pauli wrote a paperin which they proved the non-existence of such solutions. The introductioncontains the following statement:
When one tries to find a unified theory of the gravitational andelectromagnetic fields, he cannot help feeling that there is sometruth in Kaluza’s five-dimensional theory. ([14], p. 131)
46 It came out too late to be considered in the first edition of Pauli’s Relativity article. Buthe devoted to it a comparatively large space in his Supplementary Notes written in early1956 for the first English edition ([54], Suppl. Note 23, pp. 227-232; [58], pp. 276-282)
33
In fact, Einstein and Pauli offered a proof for the more general situation withan arbitrary number of additional space dimensions, fulfilling the generalisedKaluza-Klein “cylinder-condition” that the gravitational field should not de-pend on any of these extra directions. Note that this extra condition intro-duces non-dynamical background structures, so that of the 5-dimensionaldiffeomorphism group only those diffeomorphisms preserving this conditioncan act as symmetries, a point Pauli often emphasised as a deficiency re-garding the Kaluza-Klein approach.
Restricting attention to five dimensions, the explicitly stated hypothesesunderlying the proof were these ([14], p. 131; annotations in square bracketswithin quotations are mine):
H1 “The field is stationary (i.e the gik [the five-dimensional metric] areindependent of x4 [the time coordinate]).” Clearly, gik is also assumedto be independent of the fifth coordinate x5.
H2 “It [the field gik] is free from singularities.”
H3 “It is imbedded in a Euclidean space (of the Minkowski type), and forlarge values of r (r being the distance from the origin of the spatialcoordinate system) g44 has the asymptotic form g44 = −1+µ/r, whereµ 6= 0.” The last condition is meant to assure the non-triviality of thesolution, i.e. that there really is an attracting object at the spatialorigin. This becomes clear if one recalls that in the lowest weak-fieldand slow-motion approximation 1+g44 just corresponds to the Newto-nian gravitational potential. Unfortunately, the other statement: “Itis imbedded in a Euclidean space (of the Minkowski type)” seems am-biguous, since the solution is clearly not meant to be just (a portion of)5-dimensional flat Minkowski space. Hence the next closest reading ispresumably that the underlying five-dimensional spacetime manifoldis (diffeomorphic to) R5, with some non-flat metric of Minkowskiansignature (−,+,+,+,+).47
The elegant method of proof makes essential use of the fact that the suit-ably restricted group of spacetime diffeomorphisms (to those preserving thecylinder condition) is a symmetry group for the full set of equations in thesense of (3) of Definition 1. More precisely, two types of diffeomorphismsfrom that class are considered separately by Einstein and Pauli:
D1 Arbitrary ones in the three coordinates (x1, x2, x3) which leave invari-ant the (x4, x5) coordinates.
D2 Linear ones in the (x4, x5) coordinates, leaving invariant the(x1, x2, x3).
47 In fact, it turns out that formally the proof does not depend on whether the fifthdimension is space- or time-like, as noted by Einstein and Pauli ([14], p. 134).
34
Now, as a matter of fact, this innocent looking split introduces a fur-ther and, as it turns out, crucial restriction, over and above the hypothesesH1-H3. The point is that the split and, in particular, the set D2 of dif-feomorphisms simply do not exist unless the spacetime manifold, which inH3 was assumed to be R5, globally splits into R2 × R3 such that the firstfactor, R2, corresponds to the x4x5-planes of constant spatial coordinates(x1, x2, x3) and the second factor, R3, corresponds to the x1x2x3-spaces ofconstant coordinates (x4, x5). But this need not be the case if H1-H3 areassumed. The identity derived by Einstein and Pauli from the requirementthat transformations of the field induced by diffeomorphisms of the type D2are symmetries are absolutely crucial in proving the non-existence of regularsolutions.48
We now know that this additional restriction is essential to the non-existence result: There do exist solutions of the type envisaged that satisfyH1-H3, but violate the extra (and superfluous) splitting condition.49 Theyare called Kaluza-Klein Monopoles [65][31] and carry a gravitational massas well as a magnetic charge. It is hard to believe that Pauli as well asEinstein would not have been much impressed by those solutions, thoughpossibly with different conclusions, had they ever learned about them. It isalso conceivable that these solutions could have been found at the time, hadreal attempts been made, rather than—possibly—discouraged by Pauli’s andEinstein’s result. In fact, Kurt Godel, who was already in Princeton whenPauli visited Einstein, found his famous cosmological solution [26] in 1949by a very similar geometric insight that also first led to the Kaluza-Kleinmonopole [65].50
3.7 Irritations and psychological prejudices
One of Pauli’s major interests were discrete symmetries, in particular thetransformation of space inversion, ~x 7→ −~x, also called parity transforma-tion. Given a linear wave equation which is symmetric under the properorthochronous (i.e. including no space and time inversions) Poincare group,48 Specifically we mean their identity (13), which together with spatial regularity implies
the integral form (13a), which in turn leads directly to vanishing mass in (22-23a). (Allreferences are to their formulae in [14].)
49 The somewhat intricate topology of the Kaluza-Klein spacetime is this: The x5 coordi-nate parametrises circles which combine with the 2-spheres (polar coordinates (θ,ϕ))of constant spatial radius, r, into 3-spheres (Hopf fibration) which are parametrised by(θ,ϕ, x5), now thought of as Euler angles. The radii of these 3-spheres appropriatelyshrink to zero as r tends to zero, so that (r, θ,ϕ, x5) define, in fact, polar coordinatesof R4. Together with time, x4, we get R5 as global topology. Now the submanifolds ofconstant (x1, x2, x3) are those of constant (r, θ,ϕ) and have a topology R × S1 ratherthan R2, so that the linear transformations D2 in the x4x5 coordinates do not definediffeomorphisms of the Kaluza-Klein spacetime manifold.
50 Both use invariant metrics on 3-dimensional group manifolds, SU(2) in the KK case,SU(1, 1) in Godels case. This simplifies the calculations considerably.
35
one may ask whether it is also symmetric under space and time inversions.For this to be a well defined question one has to formulate conditions onhow these inversions interact with Poincare transformations. Let us focuson the operation of space inversion. If this operation is implementable byan operator P, it must conjugate each rotation and each time translationto their respective self, and each boost and each space translation to theirrespective inverse. This follows simply from the geometric meaning of spaceinversion. Hence, generally speaking, we need to distinguish the followingthree possible scenarios (recall the notation from Section 2.2):
(a) P acts on K and is a symmetry, i.e. leaves DΣ ⊂ K invariant;
(b) P acts on K and is no symmetry, i.e. leaves DΣ ⊂ K not invariant;
(c) P is not implementable on K.
It is clear that when one states that a certain equation is not symmetricunder P one usually addresses situation (b), though situation (c) also occurs,as we shall see.
Consider now the field of a massless spin-12 particle, that transforms
irreducibly under the proper orthochronous Poincare group. The field isthen either a two-component spinor, φA, which in the absence of interactionsobeys the so-called Weyl equation51
∂AA ′φA = 0 . (32)
Alternatively, one may also start from a four-component Dirac spinor,
ψ =
(φA
χA ′
)(33)
which carries a reducible representation of the proper orthochronousPoincare group: If φA transforms with A ∈ SL(2,C) then χA ′ transformswith (A†)−1 (being an element of the complex-conjugate dual space), sothat the space of the upper two components φA of ψ and the space of thelower two components χA ′ of ψ are separately invariant. One may theneliminate two of the four components by the so-called Majorana condition,which requires the state ψ to be identical to its charge-conjugate, ψc, where
C : ψ 7→ ψc := iγ2ψ∗ =
(χA
φA ′
). (34)
51 Here I use the standard Spinor notation where upper-case capital Latin indices refer to(components of) elements in spinor space (2-dimensional complex vector space), lowercase indices to the dual space, and primed indices to the respective complex-conjugatespaces. Indices are raised and lowered by using a (unique up to scale) SL(2,C) invariant2-form. An overbar denotes the map into the complex-conjugate vector space. Unlessstated otherwise, my conventions are those of [64].
36
Hence for a Majorana spinor one has φ = χ and the interaction-free Diracequation reads
γµ∂µψ :=√2
(0 ∂AA ′
∂A ′A 0
) (φA
φA ′
)= 0 . (35)
One can now either regard (32) or (35) as the interaction-free equation fora neutrino.
Here I wish to briefly recall a curious discussion between Pauli and Fierzon whether or not these two equations describe physically different state ofaffairs. Superficially this discussion is about a formal and, mathematicallyspeaking, rather trivial point. But, as we will see, it relates to deep-lyingpreconceptions in Pauli’s thinking about issues of symmetry. This makes itworth looking at this episode in some detail.
First note that there is an obvious bijection, β, between two-componentspinors and Majorana spinors, given by
β : φA 7→ (φA
φA ′
). (36)
Note also that the set of Majorana spinors is a priori a real52 vector space,though it has a complex structure, j, given by
j :
(φA
φA ′
)7→ (
iφA
−iφA ′
), (37)
with respect to which the bijection (36) satisfies β i = j β, where herei stands for the standard complex structure (multiplication with imaginaryunit i) in the space C of two-component spinors. However, regarded as amap between complex vector spaces, the bijection β is not linear.
Now, Pauli observed already in 1933 (see quotation below) that the Weylequation (32) is not symmetric under parity. Hence he concluded it couldnot be used to describe Nature. In fact, what is actually the case is thatparity cannot even be implemented as a linear map on the space of two-component spinors (case (c) above). This is easy to see and in fact true forany irreducible representation of the Lorentz group that stays irreducible ifrestricted to the rotation group (i.e. for purely primed or purely unprimedspinors).53
52 The reality structure on the complex vector space of Dirac spinors is provided by thecharge conjugation map.
53 As stated above, the geometric meaning of space inversion requires that the parityoperator (if existent) commutes with spatial rotations and conjugates boosts to theirinverse. The first requirement implies (via Schur’s Lemma) that it must be a multipleof the identity in any irreducible representation that stays irreducible when restricted tothe rotation subgroup, which contradicts the second requirement. Hence it cannot existin such representations, which are precisely those with only unprimed or only primedindices.
37
On the other hand, the Dirac equation is symmetric under space inver-sions. Indeed, the spinor-map corresponding to the inversion in the spatialplane perpendicular to the timelike normal n is given by
P : ψ 7→ ψp := ηnµγµ(ψ ρn) , (38)
where ρn : xµ 7→ −xµ +2nµ(nνxν) and where η is a complex number of unit
modulus, called the intrinsic parity of the particular field ψ. It is easy tosee that P is a symmetry of (35) for any η. Note that P2 = η21 so thatη ∈ 1,−1, i,−i, since for spinors one only requires P2 = ±1 (rather thanP2 = 1). It is also easy to verify that P commutes with C iff η = ±i. So ifwe assign imaginary parity to the Majorana field54, the operator P also actson the subspace of Majorana spinors. We conclude that the free Majoranaequation is parity invariant.
Hence it seems at first that the Weyl formulation and the Majoranaformulation differ since they have different symmetry properties. But thisis not true. Using the bijection (36), we can pull-back the parity map (38)to the space of two-component spinors, where it becomes (now either η = i
or η = −i)φA 7→ η
√2 nAA ′
(φA ′ ρn) , (39)
which is now an anti-linear map on the space of two-component spinors.All this was essentially pointed out to Pauli by Markus Fierz in a letter
dated February 6th 1957 ([45] Vol. IV, Part IV A, Doc. [2494], p. 171) in con-nection with Lee’s and Yang’s two-component theory of the neutrino. Fierzcorrectly concluded from this essential equivalence55 that the 2-componenttheory as such (i.e. without interactions) did not warrant the conclusion ofparity violation; only interactions could be held responsible for that.
This was a relevant point in the theoretical discussion at the time, as canbe seen from the fact that there were two independent papers published inThe Physical Review shortly after Fierz’s private letter to Pauli, containingthe very same observation. The first paper was submitted on February 13thby McLennan [44], the second on March 25th by Case [9]. In fact, Serpemade this observation already in 1952 [62] and emphasised it once more in1957 [63].
One might be worried about the anti-linearity of the transformation in(39). In that respect, also following Fierz, an illuminating analogy may bementioned regarding the vacuum Maxwell equations, which can be writtenin the form
i∂t~Φ− ~∇× ~Φ = 0 , ~∇ · ~Φ = 0 , (40)
where~Φ := ~E+ i~B (41)
54 Which is also the standard choice in QFT; see e.g. [69], pp. 126,226.55 Meaning the existence of a bijection that maps all quantities of interest (states, currents,
symmetries) of one theory to the other.
38
is a complex combination of the electric and magnetic field. Both equations(40) are clearly equivalent of the full set of Maxwell’s equations. It canbe shown that spatial inversions cannot be implemented as complex-lineartransformations on the complex-valued field ~Φ.56 But, clearly, we knowthat Maxwell’s equations are parity invariant, namely if we transform theelectric field as ~E 7→ −~E ρ (‘polar’ vector-field) and the magnetic field as~B 7→ ~B ρ (‘axial’ vector-field), where ρ : (t,~x) 7→ (t,−~x). This correspondsto an anti linear symmetry of (40), given by ~Φ 7→ −~Φ ρ.
Coming back to Fierz’s (and other’s) original observation for the spinorfield, they were accepted without much ado by others. For example, in hersurvey on the neutrino in the Pauli Memorial Volume, Madame Wu statesthat “It is the interaction and the interaction only that violates parity” ([18],footnote on p. 270.). In note 25c of that paper she explicitly thanks Fierzfor “enlightening discussions” on the two-component theory of the neutrino.Clearly Fierz expected his observation to be of interest to Pauli, who hadalready in the 1933 first edition of his handbook article on wave mechanicspropagated the view that Weyl’s two-component equations are57
...not invariant under reflections (interchange of left and right)and, as a consequence, not applicable to the physical reality.
But instead, Pauli reacts with a surprising plethora of ridiculing remarks:58
Dear Mr. Fierz! Your letter from the 6th is the biggest blunderyou ever commited in your life! (Probably this afternoon youwill send a correction). Have only read the first paragraph ofyour letter which originated in the asylum and was shaking withlaughter. [...] When this letter arrives (yours I will frame!) youprobably will already know everything.
Personal irritations emerged which lasted about one week through severalexchanges of letters and a phone-call. Finally Pauli essentially conceded
56 Equations (40) are equivalent to ∂AA ′fAB = 0, where fAB is the unprimed spinor
equivalent of the tensor Fµν for the electromagnetic field strength. Parity cannot belinearly implemented on this purely unprimed spinor, for reasons already explained infootnote 53.
57 German original, full sentence: “Indessen sind diese Wellengleichungen, wie ja aus ihrerHerleitung hervorgeht, nicht invariant gegenuber Spiegelungen (Vertauschung von linksund rechts) und infolge dessen sind sie auf die physikalische Wirklichkeit nicht anwend-bar” ([56], p. 234, note 54). The conclusion concerning non applicability to the physicalreality is cancelled in the 1958 edition; cf. [56], p. 150.
58 German original: “Lieber Herr Fierz! Ihr Brief vom 6. ist der großte Bock den Sieim Laufe Ihres Lebens geschossen haben! (Wahrscheinlich kommt heute Nachmittagschon eine Berichtigung von Ihnen.) Habe nur den ersten Absatz Ihres der Anstaltentsprungenen Briefes gelesen und mich geschuttelt vor lachen. [...] Wenn dieser Briefankommt (Ihren rahme ich ein!), wissen Sie wohl schon alles!” ([45], Vol. IV, Part IVA,Doc. [2497], p. 179).
39
Fierz’s point in a long letter of February 12th 1957 that also contains firsthints at Pauli’s psychological resistances (the emphasis is Pauli’s):59
Your presentation creates in me a feeling of “formal boredom”,to which the fusillade of laughter was of a compensatory nature.
This is a curious episode and not easy to understand. Pauli’s point seems tohave been that he wanted to maintain the particle-antiparticle distinctionindependently of parity, whereas Fierz pointed out that the two-componenttheory provided no corresponding structural element: In Weyl’s form theoperations C and P simply do not exist separately, in the Majorana formP exists and C is the identity (hence not distinguishing). Psychologicallyspeaking, Pauli’s point becomes perhaps more understandable if one takesinto account the fact that since the fall of 1956 he was thinking about thequestion of lepton-charge conservation. Intuitively he had therefore takenas self-evident that opposite helicities also corresponded to the particle-antiparticle duality (cf. [45], Vol. IV, Part IV A, Doc. [2497]), even thoughthis mental association did not correspond to anything in the equations. Ina letter dated February 15th 1957 he offered the following in-depth psycho-logical explanation to Fierz (the emphases are Pauli’s):60
Well, the fusillade of laughter occurred with the expression “Ma-jorana Theory” of your first letter. After this catchword I couldnot go on reading. The immediate association with Majoranaclearly has been this: “aha, particles and antiparticles should nolonger exist, these one intends to take away from me (as onetakes away a symbol from somebody)!” This causes me anxiety.I also know that since last fall the conservation of lepton charge
59 German Original: “Ihre Darstellung erzeugt bei mir das Gefuhl der ‘formalistischenLangeweile”, zu der die Lachsalve kompensatorisch war” ([45], Vol. IV, Part IVA,Doc. [2510], p. 197).
60 German original: “Also die ‘Lachsalve’ erfolgte beim Wort ‘Majorana Theorie’ Ihresersten Briefes, ich konnte nach diesem Stichwort nicht mehr weiterlesen. Die unmittel-bare Assoziation zu Majorana war naturlich ‘aha, Teilchen und Antiteilchen soll es nichtmehr geben, die will man wir wegnehmen (wie man jemandem ein Symbol wegnimmt)!’Davor habe ich Angst. Ich weiss auch, daß mir schon seit Herbst die Erhaltung derLeptonladung in der Physik ungeheuer wichtig ist – rational betrachtet, vielleicht zuwichtig. Ich habe Angst, sie konnte sich als unrichtig herausstellen und, psychologischgesehen, ist ‘Unzufriedenheit’ ein Euphemismus fur Angst. Die CP - (≡ Majorana P+
Vertauschung von Elektron und Positron) Invarianz ist mir auch wichtig, aber wenigerwichtig als die Erhaltung der Leptonladung. Es ist sicher wahr, daß ‘mein platonischerSpiegelkomplex angestochen’ war. Teilchen und Antiteilchen sind das Symbol fur jeneallgemeine Spiegelung (wie weit sie speziell platonisch ist, dessen bin ich nicht sicher).[...] Offenbar hat der ‘Spiegelungskomplex’ bei mir etwas mit Tod und Unsterblichkeitzu tun. Daher die Angst! Ware die Beziehung zwischen dem schlafenden Spiegelbildund dem Wachenden gestort, oder waren sie gar identisch (Majorana), so gabe es,psychologisch gesprochen, weder Leben (Geburt) noch Tod.” ([45], Vol. IV, Part IVA,Doc. [2517], p. 225)
40
in physics was tremendously important to me—looked upon ra-tionally probably too important. I am anxious it could turn outto be incorrect and, psychologically speaking, “discontentedness”is a euphemism for anxiety. The CP (≡ Majorana P+ exchangebetween electron and positron) invariance is also important tome, but less so than the conservation of lepton charge. It iscertainly true that it “hit upon my Platonic mirror complex”.Particles and antiparticles are the symbol for that more generalmirroring (I am not sure to what extent it is particularly pla-tonic).[...] Mirroring is also a gnostic symbol for life and death.There light is extinguished at birth and lightened up at death.[..] Obviously, for me the “mirroring complex” has something todo with death and immortality. Hence the anxiety! If the rela-tion between the sleeping mirror image and the one awake wouldbe disturbed, or if they would even be identical (Majorana), then,psychologically speaking, there would neither be life (birth) nordeath.
Fierz later commented on that episode in a personal letter to Norbert Strau-mann, parts of which are quoted in [67].
3.8 β-Decay and related issues
3.8.1 CPT
In 1955 a collection of essays by distinguished physicists appeared to cele-brate Niels Bohr’s 70th birthdays [52]. Pauli’s contribution ([52], p. 30-51) isentitled “Exclusion Principle, Lorentz Group and Reflection of Space-Timeand Charge”, whose introduction contains the following remarks:
After a brief period of spiritual and human confusion, caused byprovisional restriction to “Anschaulichkeit”, a general agreementwas reached following the substitution of abstract mathematicalsymbols, as for instance psi, for concrete pictures. Especially theconcrete picture of rotation has been replaced by mathematicalcharacteristics of the representations of the group of rotationsin three dimensional space. This group was soon amplified tothe Lorentz group in the work of Dirac. [...] The mathemat-ical group was further amplified by including the reflections ofspace and time. [...] I believe that this paper also illustrates thefact that a rigorous mathematical formalism and epistemologicalanalysis are both indispensable in physics in a complementaryway in the sense of Niels Bohr. While I try to use the former toconnect all mentioned features of the theory with help of a richer“fullness” of plus and minus signs in an increasing “clarity”, the
41
latter makes me aware that the final “truth” on the subject is still“dwelling in the abyss”.61 ([52], p. 30-31)
In some sense this paper of Pauli’s can be seen as a follow-up to his spin-statistics paper already discussed above, the main difference being that Paulinow considers interacting fields. Pauli now assumes (1) the validity of thespin-statistics correlation for interacting fields (for which there was no proofat the time), (2) invariance under (the universal cover of) the proper or-thochronous Lorentz group SL(2,C) (as in the spin-statistics paper), and(3) locality of the interactions (i.e. involving only finitely many derivatives).Then Pauli shows that this suffices to derive the so-called CPT theorem thatstates that the combination of charge conjugation (C) and spacetime reflec-tion (PT) is a symmetry.62
At the time (1955) Pauli wrote his paper it was not known whether anyof the operations of C, P, or T would separately not be a symmetry. Thischanged when in January 1957 through the experiments of Madame Wu etal., in which explicit violations of P and C were seen in processes of beta-decay, following a suggestion that this should be checked by Lee and Yangin mid 1956 [40]. Pauli had still offered a bet that this would not happenon January 17th 1957 (the emphases are Pauli’s):63
I do not believe that God is a weak left-hander and would beprepared to bet a high amount that the experiment will show asymmetric angular distribution of the electrons (mirror symme-try). For I cannot see a logical connection between the strengthof an interaction and its mirror symmetry.
In view of this firm belief in symmetry the following is remarkable: In hisCPT paper Pauli takes great care to writes down the most general ultralo-cal (i.e. no derivatives) four-fermion interaction (for the neutron, proton,electron and neutrino), which is not P invariant. In contains 10 essentiallydifferent terms with ten coupling constants C1, · · ·C10, only the first five ofwhich are parity invariant (scalars), whereas the other five are pseudoscalars,i.e change sign under spatial inversions. Apparently this he did just for the61 Here Pauli sets the following footnote: “I refer here to Bohr’s favourite verses of Schiller:
‘Nur die Fulle fuhrt zur Klarheit / Und im Abgrund wohnt die Wahrheit”’.62 Pauli used a now outdated terminology: instead of CPT he uses SR (strong reflection),
instead of PT he uses WR (weak reflection), and instead of C he uses AC (antiparticleconjugation). Preliminary versions of the CPT theorem appeared in papers by JulianSchwinger (1951)and Gerhard Luders (1954) to which Pauli refers. Two years afterPauli’s 1955 paper Res Jost gave a very elegant proof in the framework of axiomaticquantum field theory [34].
63 German original: “Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshanderist und ware bereit hoch zu wetten, daß das Experiment symmetrische Winkelverteilungder Elektronen (Spiegelinvarianz) ergeben wird. Denn ich sehe keine logischeVerbindung von Starke einer Wechselwirkung und ihrer Spiegelinvarianz.” ([45], Vol. IV,Part IVA, Doc. [2455], p. 82)
42
sake of mathematical generality without any physical motivation, as he ex-plicitly stated in a letter to Madame Wu dated January 19th 1957 (theemphases are Pauli’s):
When I considered such formal possibilities in my paper in theBohr-Festival Volume (1955), I did not think that this could havesomething to do with Nature. I considered it merely as a mathe-matical play, and, as a matter of fact, I did not believe in it whenI read the paper of Yang and Lee. [...] What prevented me untilnow from accepting this formal possibility is the question why thisrestriction of mirroring appears only in the ‘weak’ interactions,not in the strong ones. Theoretically, I do not see any interpre-tation of this fact, which is empirically so well established. ([45],Vol. IV, Part IV A, Doc. [2460], p. 89)
Lee and Yang took this possibility more serious: In an appendix to theirpaper they also write down all ten terms for the full, parity non-invariantinteraction ([40], p. 258), without any citation of Pauli.
Pauli first learnt that the experiments by Madame Wu et al. had ledto an asymmetric angular distribution from a letter by John Blatt fromPrinceton, dated January 15th 1957. There Blatt writes:
I don’t know whether anyone has written you as yet about thesudden death of parity. Miss Wu has done an experiment withbeta-decay of oriented Co nuclei which shows that parity is notconserved in β decay. [...] We are all rather shaken by by thedeath of our well-beloved friend, parity. ([45], Vol. IV, Part IV A,Doc. [2451], p. 74)
Pauli, too, was shocked as he stated in his famous letter to Weisskopf datedJanuary 27/28 1957 ([45], Vol. IV, Part IVA, Doc. [2476]). In that very sameletter Pauli already started speculating how symmetry could be restored byletting the constants Ci become dynamical field, scalar fields for i = 1, · · · , 5and pseudo-scalar ones for i = 6, · · · , 10:64
Let us imagine, for example, the terms with C1, · · · , C5 beingmultiplied with a scalar field φ(x), the terms C6, · · · , C10 mul-tiplied with a pseudo-scalar field φ(x). For God Himself, Whocan change the sign of φ(x), such a theory would be left-right-invariant—not for us mortal men, however, who do not know
64 German original: “Denken wir uns z.B. die Terme mit C1, · · · , C5 mit einem Skalarfeldφ(x), die Terme mit C6, · · · , C10 mit einem Pseudo-Skalarfeld φ(x) multipliziert. Furden Herrgott, der das Vorzeichen von φ(x) umdrehen kann, ware eine solche Theorienaturlich rechts-links-invariant – nicht aber fur uns sterbliche Menschen, die wir garnichts wissen uber jenes hypothetische neue Feld, außer daß es praktisch auf der Erderaum-zeitlich konstant (statisch-homogen) ist, und die wir noch kein Mittel haben, eszu andern.” ([45], Vol. IV, Part IVA, Doc. [2476], pp. 122-123)
43
anything about that new hypothetical field, except that it is prac-tically constant in space and time on earth (static-homogeneous),and that we do not yet65 have any means to change it.
The mechanism envisaged here to restore symmetry is just that discussed inSection 2.2, where non-dynamical backgrounds structures, Σ, are (formally)turned into dynamical quantities, Φ.
3.8.2 The Pauli group
As already mentioned, since fall of 1956 Pauli’s thinking about beta-decaywas dominated by the lepton-charge conservation. In a paper submittedon March 14th 1957, entitled On the Conservation of Lepton Charge ([38],Vol. 2, pp. 1338-1349), Pauli once more showed his mastery of symmetryconsiderations while keeping everything at the largest possible degree ofgenerality.
He starts by considering the most general ultralocal four-fermion inter-actions (not necessarily preserving parity or lepton charge) in which theneutrino field is represented by a Dirac 4-spinor, ψ. For what follows it isconvenient to think of the four components of ψ as comprising the followingfour particle states (per momentum): a left-handed neutrino, ψL, a right-handed neutrino, ψR, and their antiparticles ψc
L and ψcR respectively. Note
that this means ψcL,R := (ψL,R)c and that accordingly ψc
L is right- and ψcR is
left-handed. Here we follow the convention of [35].Next Pauli considers a four-parameter group of canonical transforma-
tions (i.e. they leave the anticommutation relations between the fermionfields invariant) of the neutrino field, henceforth called the Pauli group,whose interpretation will be given below. These transformations define asymmetry of the interaction-free equations of motions (assuming a masslessneutrino throughout), but will generally not define a symmetry once theinteraction is taken into account. Rather, the following is true (cf. [48]):Suppose that the general interaction depends on a finite number of couplingconstants ci for i = 1, · · · , n and that the equations of motion follow froman action principle with Lagrange density L Σ | Φ , where Σ represents thearray of coupling constants (we notationally ignore other non-dynamicalstructures here) and Φ the dynamical fields. Then the Pauli group acts ascovariance in a slightly stronger sense than (5), namely so that
L g · Σ | g ·Φ = L Σ | Φ . (42)
This means that on the level of the Lagrange density (or the Hamiltonian),and hence in particular at the level of the equations of motion, the transfor-mation of the dynamical fields can be compensated for by a transformation65 The “yet” is incorrectly omitted in the official translation ([45], Vol. IV, Part IVA,
p. 126).
44
of the coupling constants. A large part of Pauli’s paper is actually devotedto the determination of that compensating action of the Pauli group on thearray of coupling constants.
Next suppose the initial state is chosen to be invariant under the Pauligroup, i.e. g · Φ = Φ for all g. Then (42) implies that its evolution withinteraction parametrised by Σ (the array of ci’s) is identical to the evolutionparametrised by g ·Σ for any g. Hence the outcome of the evolution can onlydepend on the ci’s through their Pauli-invariant combinations.66 In partic-ular, since the neutrinoless double beta-decay simply has no initial neu-trino, this reasoning can be applied to it. If this lepton-charge-conservationviolating process is deemed impossible, the corresponding Pauli-invariantcombination of coupling constants to which the scattering probability isproportional67 must vanish. This, in turn, gives the sought-after constrainton the possible four-fermion interaction. For (massless) neutrinos in Majo-rana representation Pauli finally arrived at the result that either only theleft- or the right-handed component enters the interaction. It should beadded that this clever sort of reasoning was shortly before used by Purseyin a less general setting [61] in which the interaction was specialised a pri-ori to conserve lepton charge.68 More on the history of the search for theright form of the four-fermion interaction may be found in [67]. It shouldalso be mentioned that the possibility of neutrinoless double beta-decays iscurrently still under active experimental investigation at the National GranSasso Laboratory, where the 2003-2005 CUORICINO experiment set upperbounds for the Majorana mass of the electron neutrino well below one eV.The upcoming next-generation experiment, CUORE, is designed to lowerthis bound to 5 · 10−2meV ; compare [30].
What is the interpretation of the Pauli group? Mathematically it isisomorphic to U(2), the group of 2 × 2 unitary matrices acting on a two-dimensional complex vector space. Here there are two such spaces (per 4-momentum) in which it acts: the ‘left-handed subspace’ that is spanned bythe two left-handed components ψL and ψc
R, and the ‘right-handed subspace’that is spanned by the two right-handed components ψR and ψc
L. The two66 For illustrative purposes we argue here as if all fields were classical and obeyed classical
equations of motion, though Pauli clearly considers the quantum theory where thefields become operators. The principal argument is the same, though what makes a bigdifference between the classical and the quantum case is that in the latter we can moreeasily ascertain the existence of invariant initial states. This is because in quantumtheory, assuming there are no superselection rules at work, the superposition principlealways allows us to construct invariant initial states by group-averaging any given stateover the group (which is here compact, so that the averaging is unambiguously defined).Such states would, for example, appropriately represent physical situations where thoseobservables that distinguish between the states in the group orbit are not measured,may it be for reasons of practice or of principle.
67 It will be a quadratic combination in leading order of perturbation theory. Explicitcalculations had been done by Pauli’s assistant Charles Enz [15].
68 In terms of the Pauli group, Pursey did not consider the U(1) part.
45
actions of U(2) in these spaces are complex conjugate to each other (seeequation (43)). Usually one thinks of the Pauli group as U(1) × SU(2),which is a double cover of U(2), so that the four real parameters are writtenas a phase exp(iα), parametrising U(1), and two complex parameters a, bsatisfying |a|2 + |b|2 = 1, which give three real parameters when split intoreal and imaginary part and which parametrise a 3-sphere that underliesSU(2) as group manifold. In this parametrisation the action of the Pauligroup reads (an asterisk stands for complex conjugation):69(
ψL
ψcR
)7→ exp
(+iα
) (a b
−b∗ a∗
) (ψL
ψcR
), (43a)(
ψcL
ψR
)7→ exp
(−iα
) (a∗ b∗
−b a
) (ψc
L
ψR
). (43b)
Invariance under the Pauli group is now seen to correspond to an am-biguity in the particle-antiparticle distinction. This ambiguity would onlybe lifted by interactions that allowed to distinguish the two left and thetwo right states respectively. In the absence of such interactions the variousdefinitions of ‘particle’ and ‘antiparticle’ are physically indistinguishable, sothat the Pauli group acts by gauge symmetries in the sense of Section 2.3.
Also, the different presentations of the two-component theory, alreadydiscussed in Section 3.7 can be seen here. The Majorana condition readsψ = ψc, which in terms of the four components introduced above leads toψL = ψc
R and ψR = ψcL. This can be read in two different ways, depending
on whether one addresses ψL, ψcL or ψL, ψR as independent basic states. In
the first case one would say that there is a left-handed neutrino and itsright-handed antiparticle, whereas in the second case one regards the tuple(ψL, ψR) as respectively the left- and right-handed components of a singleparticle which is identical to its antiparticle.
Beyond weak interaction and beta-decay, the Pauli group played a veryimportant role in Pauli’s brief participation in Heisenberg’s programme fora unified field theory. It was Pauli who first showed that the (so far classi-cal) non-linear spinor equation proposed by Heisenberg was invariant underthe Pauli group (cf. Heisenberg’s account in his letter to Zimmermann fromJan. 7th 1958 in [45], Vol. IV, Part IVB, p. 779). In this new context the U(1)
part of the Pauli group was connected to conservation of baryon charge andthe SU(2) part acquired the meaning of isospin symmetry.70 The central69 Usually the Pauli group is written in terms of the 4-component neutrino field ψ asψ 7→ exp(iαγ5)(aψ + bγ5ψ
c), where ψc := iγ2ψ∗ is the charge conjugate field. But
this is easily seen to be equivalent to (43) if one sets ψL,R = 12(1 ± γ5)ψ and ψc
R,L =12(1 ± γ5)ψc. The more explicit form (43) is better suited for the interpretational
discussion; cf. [35]. The two-to-one homomorphism from U(1)× SU(2) to U(2) is givenby
`exp(iα) , A
´7→ exp(iα)A whose kernel is
(1,1) , (−1,−1)
.
70 The non-linear spinor equation was at that stage not designed to include weak interac-tion.
46
importance of isospin for this programme may already be inferred from thetitle of the proposed common publication by Heisenberg and Pauli, whichreads: On the Isospin Group in the Theory of Elementary particles. How-ever, due to Pauli’s later retreat from this programme, the manuscript (cf.[45], Vol. IV, Part IV B, pp. 849-861) for this publication never grew beyondthe stage of a preprint.
3.8.3 Cosmological speculations
In his last paper on the subject of discrete symmetries, entitled The Vio-lation of Mirror-Symmetries in the Laws of Atomic Physics71 ([38], Vol. 2,pp. 1368-1372), Pauli comes back to the question which bothered him most:How is the strength of an interaction related to its symmetry properties?He says that having established a violation of C and P symmetry for weakinteractions, we may ask why they are maintained for strong and electro-magnetic interactions, and whether the reason for this is to be found inparticular properties of these interactions. He ends with some speculationson possible connections between violations of C and P symmetry in the lawsof microphysics on one hand, and properties of theories of gravitation andits cosmological solutions on the other:72
Second, one can try to find and justify a connection betweensymmetry violation in the small with properties of the Universein the large. But this exceeds the capabilities of the presentlyknown theory of gravity. [...] New ideas are missing to go be-yond vague speculations. But this shall not be taken as definiteexpression for the impossibility of such a connection.
It may be of interest to contrast this expression of a certain open-mindednessfor speculations concerning the physics of elementary particles on one sideand large-scale cosmology on the other, with a more critical attitude fromPauli’s very early writings. In Section 65 of his Relativity article, wherePauli discussed Weyl’s attempt for a unifying theory of gravity and electro-magnetism (to which Pauli himself actively contributed), he observes thatin Weyl’s theory (as well as in Einstein’s own attempts from that time) itis natural to suspect a relation between the size of the electron and the size71 German original: “Die Verletzung von Spiegelungs-Symmetrien in den Gesetzen der
Atomphysik”.72 German original: “Zweitens kann man versuchen, einen Zusammenhang der Symme-
trieverletzungen in Kleinen mit Eigenschaften des Universums im Grossen aufzufindenund zu begrunden. Dies uberschreitet aber die Moglichkeiten der jetzt bekannten The-orien der Gravitation. [...] Um bei der Frage des Zusammenhangs zwischen demKleinen und dem Grossen uber vage Spekulationen hinauszugelangen, fehlen daher nochwesentlich neue Ideen. Hiermit soll jedoch nicht die Unmoglichkeit eines solchen Zusam-menhanges bestimmt behauptet werden.” ([38], Vol. 2, p. 1371)
47
(mean curvature radius) of the universe. But then he comments somewhatdismissively that this might seem somewhat fantastic73([54], p. 202).
4 Conclusion
I have tried to display some of the aspects of the notion of symmetry inthe work of Wolfgang Pauli which to me seem sufficiently interesting intheir own right. In doing this I have drawn freely from Pauli’s scientificœvre, irrespectively of whether the particular part is commonly regardedas established part of present-day scientific knowledge or not. Pauli’s faithin the explanatory power of symmetry principles clearly shows up in allcorner of his œvre, but it also appears clearly rooted beyond the limits ofhis science.
In the editorial epilogue to the monumental collection of Pauli’s scientificcorrespondence, Karl von Meyenn reports that many physicists he talked toat the outset of his project spoke against the publication of those letters thatcontained ideas which did not stand the test of time ([45], Vol. IV, Part IV B,p. 1375). Leaving aside that this must clearly sound outrageous to the his-torian, it is, in my opinion, also totally misguided as far as the scientificendeavour is concerned. Science is not only driven by the urge to know butalso, and perhaps most importantly, by the urge to understand. No one whoas ever actively participated in science can deny that. One central aspectof scientific understanding, next to offering as many as possible alternativeand complementary explanations for the actual occurrences in Nature, is tocomprehend why things could not be different from what they appear to be.The insight into a theoretical or an explanatory failure can be as fruitful asan experimental failure. What makes Pauli a great scientist, amongst theother most obvious reasons, is not that he did not err—such mortals clearlydo not exist—, but that we can still learn much from where he erred andhow he erred. In that sense, let me end by the following words from JohannWolfgang von Goethe’s Maximen und Reflexionen (# 1292):
Wenn weise Manner nicht irrten, mußten die Narren verzweifeln.
(If wise men did not err, fools should despair.)
73 German original: “...was immerhin etwas phantastisch erscheinen mag.” ([58], p. 249)
48
Acknowledgements: I sincerely thank Harald Atmanspacher and HansPrimas for the invitation to talk at the conference on Wolfgang Pauli’s Philo-sophical Ideas and Contemporary Science on the Monte Verita in Ascona,Switzerland. I also thank Norbert Straumann for comments and suggestionsfor improvement. Finally I wish to express my strongest appreciation to theeditor of Pauli’s Scientific Correspondence, Karl von Meyenn, without whoseadmirable editorial work we would not be in the position to share many ofWolfgang Pauli’s wonderful insights. Thank you!
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